Comparing the Radiopharmaceutical Drug, [177Lu]Lu-DOTATATE, to Standard of Care Treatment for Patients With Meningioma That Has Come Back After Prior Treatment

Inclusion Criteria: STEP 1 REGISTRATION * Aged \>= 18 years * Histologically confirmed diagnosis of WHO grade 1-3 meningioma * Disease progression following at least one prior surgical intervention (biopsy or resection) and at least one prior course of radiation. * The patient is not considered a candidate for additional surgery and/or radiation or the patient has declined additional surgery and/or radiation. * Presence of measurable contrast-enhancing disease on gadolinium-enhanced MRI brain scan defined as at least one lesion with two perpendicular diameters measuring ≥10 mm on two or more axial slices (≤ 5 mm interslice thickness, ≤ 1 mm interslice gap) per current RANO meningioma criteria * Progression of disease determined by local radiology review per current RANO meningioma criteria, defined as: * ≥ 15% increase in sum of product of perpendicular measurements of up to 5 measurable target lesions within the last 6 months (196 days), or * ≥ 25% increase in sum of product of perpendicular measurements of up to 5 measurable target lesions within the last 12 months (379 days), or * Development of a new measurable lesion * The following scans must be available for submission for central radiology review: * Pre-progression gadolinium-enhanced MRI brain scan no older than 12 months (379 days) that serves as a reference for evaluation of radiographic disease progression * Progression gadolinium-enhanced MRI brain scan STEP 2 REGISTRATION * Progression of disease determined by central radiology review per current RANO meningioma criteria, defined as: * ≥ 15% increase in sum of product of perpendicular measurements of up to 5 measurable target lesions within the last 6 months (196 days), or * ≥ 25% increase in sum of product of perpendicular measurements of up to 5 measurable target lesions within the last 12 months (379 days), or * Development of a new measurable lesion. * \[68Ga\]Ga-DOTATATE uptake on PET-CT. Positive uptake is defined as uptake at least as high as liver, based on the uptake in at least one target lesion. * Both the patient and investigator must agree that the patient will NOT receive SSTR2-targeted therapy, surgical resection, or radiation therapy until progression of disease while on study treatment. * Patients must be willing and able to undergo regular MRI scans of the brain and \[68Ga\]Ga-DOTATATE PET-CT imaging during the study. * Patients must have recovered to CTCAE grade ≤1 or pretreatment baseline from clinically significant adverse events related to prior therapy (exclusions include alopecia, lymphopenia, sensory neuropathy ≤ grade 2, or other ≤ grade 2 not constituting a safety risk based on the investigator's judgment). * Adequate organ and bone marrow function as defined below (within 28 days prior to step 2 registration): * Absolute neutrophil count (ANC) ≥ 1500/mm3 * Platelet count ≥ 75,000/mm3 * Hemoglobin ≥ 8 g/dL * Creatinine clearance (calculated by the Cockroft-Gault method) ≥40mL/min * AST (SGOT) and ALT (SGPT) ≤ 3 x the laboratory upper limit of normal (ULN) * Total serum bilirubin ≤ 3 x ULN (except participants with Gilbert's Syndrome, who can have a total bilirubin ≤ 5 x ULN) * Potassium within normal limits. Exclusion Criteria: * Patients with a clinical diagnosis of NF2-related schwannomatosis or with a known molecular diagnosis of NF2-related schwannomatosis. * Patients with radiation-associated meningiomas. * Patients with known intraspinal meningiomas or meningioma metastases outside the skull/spinal column. Meningiomas invading the skull base or orbits are not excluded. * Prior SSTR2-targeted therapy, e.g. Somatostatin LAR or short-acting Octreotide. No limit on number of prior non-SSTR2-tartgeted systemic therapies. * Unstable neurological symptoms requiring steroids to control symptoms at a dose of \>2 mg of dexamethasone (or equivalent) daily within 28 days prior to step 2 registration. * Patients requiring immediate local therapy (e.g. surgical resection). * Surgical procedure within the timeframes listed below, prior to step 2 registration. * 28 days from any prior craniotomy * 7 days from stereotactic biopsy Note: There is no limit to the number of prior surgical interventions * Treatment within the timeframes specified below, prior to step 2 registration. * 28 days (or 5 half-lives, whichever is longer) for cytotoxic chemotherapy, biologic agent, investigational agent or any other systemic agent prescribed for the purpose of treating meningioma * 6 weeks from nitrosoureas Note: There is no limit to the number of prior systemically administered therapeutic agents. * A target lesion is excluded if it has received any of the following: * More than 2 total prior courses of radiation * More than 1 prior course of standard fractionated external beam radiation therapy (EBRT) * Radiation treatment to the target lesion completed \<24 weeks (168 days) before Step 2 registration. For purposes of this criterion, one course of EBRT counts as one course regardless of duration, and each course of stereotactic radiation (1-5 fractions) counts as one course. Accordingly, a target lesion may have received at most one prior course of EBRT plus one prior course of stereotactic radiation, or two prior courses of stereotactic radiation. Prior radiation history must be evaluated on a lesion-by-lesion basis, including radiation delivered before any subsequent surgical resection. * All types of radioligand therapy at any time prior to registration. Radioligands received for diagnostic purposes are not exclusionary. * Known hypersensitivity to somatostatin analogues or any component of the \[68Ga\]Ga- DOTATATE or \[177Lu\]Lu-DOTATATE formulations. * Active infection requiring current use of intravenous therapy with antibiotics. * Active cardiovascular disease: cerebral vascular accident/stroke (≤ 6 months prior to registration), myocardial infarction (≤ 6 months prior to registration), congestive heart failure (≥ NYHA class II), unstable angina pectoris, or serious cardiac arrhythmia requiring medication. * An active malignancy ≤ 3 years. Note: Patients with a malignancy whose natural history or treatment does not have the potential to interfere with the safety or efficacy assessment of the investigational regimen are eligible for this trial. * Pregnant and/or breastfeeding patients who are unwilling to discontinue breast feeding. * Participants of childbearing potential must have a negative pregnancy test within 14 days of study entry.