Fulvestrant and Palbociclib With or Without Copanlisib in Treating Patients With Hormone Receptor Positive, HER2 Negative, Stage IV Breast Cancer

Inclusion Criteria: * Patients must have histologically confirmed estrogen receptor (ER) and/or progesterone receptor (PR) positive, HER2 negative or non-amplified breast cancer that is stage IV, with measurable or non-measurable disease; ER/PR positivity is defined as at least 1% positive or Allred score of at least 3 * All patients must agree to provide archival tumor material for research and must agree to undergo research tumor biopsy before treatment if presence of easily accessible lesions (judged by the treating physician); for patients with bone only disease, or patients without easily accessible lesions for the baseline research biopsy, availability of archival tumor material (2 x 4-5 micron section unstained slides, plus 15-20 x 10 micron section unstained slides or a tumor rich block) from previous breast cancer diagnosis or treatment is required for PTEN and PIK3CA analysis * No more than 1 prior chemotherapy in the metastatic setting; there is no limit on prior lines of endocrine therapy; (for patients enrolling to the safety run-in portion of the study, prior fulvestrant, CDK4/6 inhibitor, and everolimus is allowed) * For patients enrolling to the randomized phase II portion of this study, demonstrated resistance to prior endocrine therapy in the metastatic setting is required; this is defined as: * Progressed on prior endocrine therapy in the metastatic setting or * Relapsed on adjuvant endocrine therapy or * Relapsed within 12 months of completing adjuvant endocrine therapy or * If received adjuvant CDK4/6 inhibitor, relapsed at least 2 years after completion of adjuvant CDK4/6 inhibitor * Washout from prior systemic anti-cancer therapy of at least 3 weeks or 5 half-lives of the drug before the start of study treatment; washout from prior radiation therapy of at least 2 weeks before the start of the study treatment * Eastern Cooperative Oncology Group (ECOG) performance status =\< 2 (Karnofsky \>= 60%) * Absolute neutrophil count \>= 1,500/mcL collected no more than 7 days before starting study treatment * Platelets \>= 75,000/mcL collected no more than 7 days before starting study treatment * Hemoglobin \>= 8.0 g/dL collected no more than 7 days before starting study treatment * Total bilirubin =\< 1.5 x institutional upper limit of normal (=\< 3 x institutional upper limit of normal for patients with Gilbert syndrome) collected no more than 7 days before starting study treatment * Aspartate aminotransferase (AST) (serum glutamic oxaloacetic transaminase \[SGOT\])/alanine aminotransferase (ALT) (serum glutamic pyruvic transaminase \[SGPT\]) =\< 2.5 x institutional upper limit of normal (=\< 5 x institutional upper limit of normal for patients with liver involvement) collected no more than 7 days before starting study treatment * Glomerular filtration rate \>= 40 mL/min/1.73 m\^2 according to the Modification of Diet in Renal Disease (MDRD) abbreviated formula collected no more than 7 days before starting study treatment * Lipase =\< 1.5 x ULN collected no more than 7 days before starting study treatment * International normalized ratio (INR) and partial thromboplastin time (PTT) =\< 1.5 x ULN collected no more than 7 days before starting study treatment * Glycosylated hemoglobin (HbA1c) =\< 8.5% collected no more than 7 days before starting study treatment * Left ventricular ejection fraction (LVEF) \>= 50% * Patients who are therapeutically treated with an agent such as warfarin or heparin will be allowed to participate provided that their medication dose and INR/PTT is stable * Prophylactic antiemetics may be administered according to standard practice; the routine use of standard antiemetics, including 5-HT3 blockers, such as granisetron, ondansetron, or an equivalent agent, is allowed as needed * Patients may be postmenopausal or premenopausal women * Women of child-bearing potential and men must agree to use adequate contraception (hormonal or barrier method of birth control; abstinence) prior to study entry, for the duration of study participation, and for 6 months after the last dose of copanlisib; should a woman become pregnant or suspect she is pregnant while she or her partner is participating in this study, she should inform her treating physician immediately; men treated or enrolled on this protocol must also agree to use adequate contraception prior to the study, for the duration of study participation, and 6 months after completion of copanlisib administration * For patients enrolling to the safety run-in portion of the study, a history of treated brain metastases is allowed if there is no disease progression symptomatically and by imaging within 28 days prior to registration and if the patient is off steroids * Ability to understand and willing to sign a written informed consent document Exclusion Criteria: * For patients enrolling to the randomized phase II portion of the study, prior treatment with a CDK4/6 inhibitor or fulvestrant, or a PI3K inhibitor in the metastatic setting is not allowed * Patients who have had chemotherapy or radiotherapy within 4 weeks (6 weeks for nitrosoureas or mitomycin C) prior to entering the study * Patients who have not recovered from adverse events due to prior anti-cancer therapy (i.e., have residual toxicities \> grade 1 with the exception of alopecia) * Immunosuppressive therapy is not allowed while on study * Patients who are receiving any other investigational agents * Receiving anti-arrhythmic therapy (beta blockers or digoxin are permitted) * Patients with brain metastasis are not eligible for the randomized phase II portion of the study; for the safety run-in portion of the study, patients with progressive brain metastases should be excluded; for patients enrolling to the randomized phase II portion of the study, patients with a history of brain metastases should be excluded from this clinical trial * History of allergic reactions attributed to compounds of similar chemical or biologic composition to copanlisib, PI3K inhibitors, or other agents used in study * The concomitant use of strong inhibitors of CYP3A4 (e.g., ketoconazole, itraconazole, clarithromycin, ritonavir, indinavir, nelfinavir and saquinavir), and strong inducers of CYP3A4 (e.g. rifampin, phenytoin, carbamazepine, phenobarbital, St. John's wort) are not permitted from 14 days prior to enrollment until the end of the study; other medications that are prohibited while on copanlisib treatment: * Herbal medications/preparations (except for vitamins) * Anti-arrhythmic therapy other than beta blockers or digoxin * Systemic corticosteroid therapy at a daily dose higher than 15 mg prednisone or equivalent is not permitted while on study; previous corticosteroid therapy must be stopped or reduced to the allowed dose at least 7 days prior to the computed tomography (CT)/magnetic resonance imaging (MRI) screening; if a patient is on chronic corticosteroid therapy, corticosteroids should be de-escalated to the maximum allowed dose before the screening; patients may be using topical or inhaled corticosteroids; short-term (up to 7 days) systemic corticosteroids above 15 mg prednisolone or equivalent will be allowed for the management of acute conditions (e.g., treatment non-infectious pneumonitis); the use of corticosteroids as antiemetics prior to copanlisib administration will not be allowed * Major surgical procedure or significant traumatic injury (as judged by the investigator) within 28 days before start of treatment, or have not recovered from major side effects, open biopsy within 7 days before start of treatment * Uncontrolled intercurrent illness, including but not limited to, symptomatic congestive heart failure (\> New York Heart Association \[NYHA\] class 2), unstable angina pectoris, new-onset angina, cardiac arrhythmia, uncontrolled hypertension despite optimal medical management, seizure disorder requiring medication, or psychiatric illness/social situations that would limit compliance with study requirements * Proteinuria \>= grade 3 as assessed by a 24 hour \[h\] protein quantification or estimated by urine protein: creatinine ratio \> 3.5 on a random urine sample * Myocardial infarction \< 6 months before start of treatment * History of bleeding diathesis; any hemorrhage or bleeding event \>= grade 3 within 4 weeks prior to the start of study medication * History or concurrent condition of interstitial lung disease of any severity and/or severely impaired lung function * History of having received an allogeneic bone marrow or organ transplant * Patients with non-healing wound, ulcer, or bone fracture * Patients with active, clinically serious infections \> grade 2 (Common Terminology Criteria for Adverse Events \[CTCAE\] version \[v\]4.0) * Patients with uncontrolled type I or II diabetes mellitus; uncontrolled diabetes is defined as HbA1c \> 8.5% * Arterial or venous thrombotic or embolic events such as cerebrovascular accident (including transient ischemic attacks), deep vein thrombosis or pulmonary embolism within 3 months before the start of study medication * Concurrent diagnosis of pheochromocytoma * Has undergone blood or platelet transfusion \< 7 days prior to start of treatment * Pregnant women are excluded from this study; breastfeeding should be discontinued if the mother is treated with copanlisib; these potential risks may also apply to other agents used in this study * Hepatitis B (HBV) or hepatitis C (HCV); all patients must be screened for HBV and HCV up to 28 days prior to study drug start using the routine hepatitis virus lab panel; patients positive for hepatitis B surface antigen (HBsAg) and/or hepatitis B core antibody (HBcAb) will be eligible if they are negative for HBV DNA, these patients should receive prophylactic antiviral therapy; patients positive for anti-HCV antibody will be eligible if they are negative for HCV ribonucleic acid (RNA) * Human immunodeficiency virus (HIV)-positive patients on combination antiretroviral therapy are ineligible; appropriate studies will be undertaken in patients receiving combination antiretroviral therapy when indicated * Cytomegalovirus (CMV) polymerase chain reaction (PCR) positive at baseline * Patients with history of, or current autoimmune disease are not eligible