Inclusion Criteria:  Documented informed consent of the participant and/or legally authorized representative  Assent, when appropriate, will be obtained per institutional guidelines  Agreement to allow the use of archival tissue from diagnostic tumor biopsies  If unavailable, exceptions may be granted with study principal investigator (PI) approval Age: >= 18 years Eastern Cooperative Oncology Group (ECOG) =< 2 Life expectancy >= 3 months (per physician assessment) Ability to take oral medication  Pathologically confirmed MCL, with documentation of monoclonal B cells showing one of the following:  Overexpression of cyclin D1 in association with other relevant markers (e.g., CD19, CD20, PAX5, CD5), OR Chromosomal translocation t(11;14)(q13; q32), as assessed by cytogenetics, fluorescent in situ hybridization (FISH), or polymerase chain reaction (PCR) Documented failure to achieve at least partial response (PR) with, or documented disease progression after the most recent treatment regimen At least 1 prior treatment regimen for MCL which either included chemo-immunotherapy or a targeted agent (i.e., ibrutinib) administered for at least 2 cycles Have radiologically measurable lymphadenopathy or extranodal lesion, (defined as >= 1 lesion that measures >= 2.0 cm in the longest diameter), or splenomegaly, or bone marrow involvement with or without malignant lymphocytosis Absolute neutrophil count (ANC) >= 1,000/mm^3 without bone marrow involvement or ANC >= 500/mm^3 with bone marrow involvement (to be performed within 30 days prior to day 1 of protocol therapy unless otherwise stated)  Platelets >= 50,000/mm^3 without bone marrow involvement or platelets >= 30,000/mm^3 with bone marrow involvement (to be performed within 30 days prior to day 1 of protocol therapy unless otherwise stated)  NOTE: Platelet transfusions are not permitted within 7 days of platelet assessment Total bilirubin =< 2 x upper limit of normal (ULN) (unless has Gilbert's disease or documented liver/biliary involvement by the lymphoma) (to be performed within 30 days prior to day 1 of protocol therapy unless otherwise stated) Aspartate aminotransferase (AST) =< 2.5 x ULN (to be performed within 30 days prior to day 1 of protocol therapy unless otherwise stated) Alanine aminotransferase (ALT) =< 2.5 x ULN (to be performed within 30 days prior to day 1 of protocol therapy unless otherwise stated) Creatinine clearance of >= 30 mL/min per 24 hour urine test or the Cockcroft-Gault formula (to be performed within 30 days prior to day 1 of protocol therapy unless otherwise stated) If not receiving anticoagulants: International normalized ratio (INR) OR prothrombin time (PT) =< 1.5 x ULN (to be performed within 30 days prior to day 1 of protocol therapy unless otherwise stated) If on anticoagulant therapy: PT must be within therapeutic range of intended use of anticoagulants (to be performed within 30 days prior to day 1 of protocol therapy unless otherwise stated) If not receiving anticoagulants: Activated partial thromboplastin time (aPTT) =< 1.5 x ULN (to be performed within 30 days prior to day 1 of protocol therapy unless otherwise stated) If on anticoagulant therapy: aPTT must be within therapeutic range of intended use of anticoagulants (to be performed within 30 days prior to day 1 of protocol therapy unless otherwise stated)  Left ventricular ejection fraction (LVEF) >= 45% (to be performed within 30 days prior to day 1 of protocol therapy unless otherwise stated)  Note: To be performed within 28 days prior to day 1 of protocol therapy  Seronegative for human immunodeficiency virus (HIV) antigen (Ag)/antibody (Ab) combo, hepatitis C virus (HCV), active hepatitis B virus (HBV) (surface antigen negative) (to be performed within 30 days prior to day 1 of protocol therapy unless otherwise stated)  If positive, hepatitis C ribonucleic acid (RNA) quantitation must be performed  Meets other institutional and federal requirements for infectious disease titer requirements (to be performed within 30 days prior to day 1 of protocol therapy unless otherwise stated)  Note Infectious disease testing to be performed within 28 days prior to day 1 of protocol therapy Lipase =< 1.5 ULN (to be performed within 30 days prior to day 1 of protocol therapy unless otherwise stated) Glycosylated hemoglobin (HbA1c) =< 8.5% (to be performed within 30 days prior to day 1 of protocol therapy unless otherwise stated)  Women of childbearing potential (WOCBP): negative urine or serum pregnancy test (to be performed within 30 days prior to day 1 of protocol therapy unless otherwise stated)  If the urine test is positive or cannot be confirmed as negative, a serum pregnancy test will be required  Agreement by females and males of childbearing potential to use an effective method of birth control or abstain from heterosexual activity for the course of the study through at least 30 days after the last dose of protocol therapy  Childbearing potential defined as not being surgically sterilized (men and women) or have not been free from menses for > 1 year (women only) without an alternative medical cause.  Exclusion Criteria:  Concurrent enrollment in another therapeutic investigational study Prior treatment with venetoclax (or other investigational small molecule BCL2 inhibitors) or with copanlisib Prior allogeneic stem cell transplant  Prior therapeutic intervention with any of the following:  Therapeutic anticancer antibodies within 2 weeks Radio- or toxin-immunoconjugates within 10 weeks All other chemotherapy, radiation therapy within 30 days prior to initiation of therapy Targeted therapy within 6 half-lives Vaccinated with live vaccines within 4 weeks of the first dose of study drug Systemic continuous corticosteroid therapy at a daily dose higher than 20 mg prednisone or equivalent is not allowed. Participants may be using topical or inhaled corticosteroids Concurrent administration of medications or food that are strong inhibitors or inducers of CYP3A4 taken within 7 days of starting study treatment Current evidence of central nervous system involvement by the lymphoma Uncontrolled active systemic infection Unresolved toxicities (except alopecia) from prior anticancer therapy (including radiation) that have not resolved to grade =< 1 (per Common Terminology Criteria for Adverse Events [CTCAE] version [v] 5.0), or to the levels dictated in this protocol  Known history of immunodeficiency virus (HIV) or hepatitis B or hepatitis C infection  Participants who are positive for hepatitis B core antibody, hepatitis B surface antigen (HBsAg), or hepatitis C antibody must have a negative polymerase chain reaction (PCR) result before enrollment. Those who are PCR positive will be excluded  History of prior malignancy except:  Malignancy treated with curative intent and no known active disease present for >= 2 years prior to initiation of therapy on current study Adequately treated non-melanoma skin cancer or lentigo maligna (melanoma in situ) without evidence of disease Adequately treated in situ carcinomas (e.g., breast, cervical, esophageal, etc.) without evidence of disease Asymptomatic prostate cancer managed with "watch and wait" strategy or hormonal therapy Major surgery within 4 weeks of the first dose of study drug Currently active, clinically significant cardiovascular disease, such as uncontrolled arrhythmia or class 3 or 4 congestive heart failure as defined by the New York Heart Association Functional classification; or a history of myocardial infarction, unstable angina, or acute coronary syndrome within 6 months of screening Uncontrolled arterial hypertension despite optimal medical management History or concurrent condition of interstitial lung disease of any severity and/or severely impaired lung function, such as patients requiring oxygen supplementation Uncontrolled diabetes mellitus despite optimal medical management (per investigator's opinion) Arterial or venous thrombotic or embolic events such as cerebrovascular accident (including transient ischemic attacks) within 3 months before the start of study medication Known hypersensitivity to any of the test drugs, test drug classes, or excipients in the formulation Females only: Pregnant or breastfeeding Any other condition that would, in the investigator's judgment, contraindicate the patient's participation in the clinical study due to safety concerns with clinical study procedures Prospective participants who, in the opinion of the investigator, may not be able to comply with all study procedures (including compliance issues related to feasibility/logistics)