Comparing Combinations of Targeted Drugs for Advanced Non-Small Cell Lung Cancer That Has EGFR and MET Gene Changes (A Lung-MAP Treatment Trial)

Inclusion Criteria: * Patients must meet all SCREENING/PRE-SCREENING and SUB-STUDY REGISTRATION COMMON ELIGIBILITY CRITERIA as specified in S1400: Phase II/III Biomarker-Driven Master Protocol for Previously Treated Squamous Cell Lung Cancer (Lung-Map) * Participants must have been assigned to S1900G by the Southwest Oncology Group (SWOG) Statistics and Data Management Center (SDMC). Assignment to S1900G is determined by the LUNGMAP protocol * Participants must have documentation of NSCLC with a sensitizing EGFR mutation and have radiologically or clinically progressed (in the opinion of the treating physician) on osimertinib, alone or in combination with other agent(s), as their most recent line of therapy. Any number of prior lines of therapy is allowed * Participants must have a MET amplification determined by tissue-based or blood-based (circulating tumor DNA \[ctDNA\]) next generation sequencing (NGS) assay. MET amplifications may have been determined based on tissue submitted for testing by Foundation Medicine Inc (FMI) through the LUNGMAP screening protocol or using test results completed outside of the study. Tissue or blood must be obtained after disease progression on osimertinib (alone or in combination with another agent\[s\]). The testing must be done within a laboratory with Clinical Laboratory Improvement Act (CLIA), International Organization for Standardization (ISO)/Independent Ethics Committee (IEC), College of American Pathologists (CAP), or similar certification * Note: Participants previously tested for and determined to have MET amplified NSCLC, at the time of progression on osimertinib, outside of LUNGMAP, must also submit tissue for central FMI testing on the LUNGMAP screening protocol, if available * Participants must have either measurable disease or non-measurable disease documented by CT or MRI. The CT from a combined PET/CT may be used to document only non-measurable disease unless it is of diagnostic quality. Measurable disease must be assessed within 28 days prior to sub-study randomization. Non-measurable disease must be assessed within 42 days prior to sub-study randomization. All known sites of disease must be assessed and documented on the Baseline Tumor Assessment Form. Participants whose only measurable disease is within a previous radiation therapy port must demonstrate clearly progressive disease (in the opinion of the treating investigator) prior to sub-study randomization to be considered measurable * Participants must have a CT with contrast or MRI scan of the brain to evaluate for central nervous system (CNS) disease within 42 days prior to sub-study randomization * Participants with symptomatic CNS metastasis (brain metastases or leptomeningeal disease) must be neurologically stable and have a stable or decreasing corticosteroid requirement for at least 5 days before sub-study randomization * Participants must have recovered (=\< grade 1) from any side effects of prior therapy, except for alopecia and vitiligo * Participants must be able to swallow tablets whole * Absolute neutrophil count \>= 1.5 x 10\^3/uL (within 28 days prior to sub-study randomization) * Hemoglobin \< 9.0 g/dL (within 28 days prior to sub-study randomization) * Platelets \>= 100 x 10\^3/uL (within 28 days prior to sub-study randomization) * Total bilirubin =\< 1.5 x institutional upper limit of normal (ULN) unless history of Gilbert's disease (within 28 days prior to sub-study randomization). Participants with history of Gilbert's disease must have total bilirubin =\< 5 x institutional ULN * Aspartate aminotransferase (AST) and alanine aminotransferase (ALT) =\< 2.5 x institutional ULN. Participants with history of liver metastasis must have AST =\< 5 x ULN (within 28 days prior to sub-study randomization) * Participants must have a serum creatinine =\< the IULN OR calculated creatinine clearance \>= 50 mL/min using the following Cockcroft-Gault Formula. This specimen must have been drawn and processed within 28 days prior to sub-study randomization * Participants' most recent Zubrod performance status must be 0-1 and be documented within 28 days prior to sub-study randomization * Participants must have an electrocardiogram (ECG) performed, with a Fridericia's Correction Formula (QTcF) =\< 470 msec, within 28 days prior to sub-study randomization. It is suggested that a local cardiologist review the QTcF intervals * Participants must have a completed medical history and physical exam within 28 days prior to sub-study randomization * Participants must have a urinalysis performed 28 days prior to sub-study randomization. Participant must have a urinary protein =\< 1+ on dipstick or routine urinalysis (UA). Random analysis of urine protein with a normal value is sufficient. If urine dipstick or routine analysis indicated proteinuria \>= 2+, then a 24-hour urine is to be collected and demonstrate \< 2000 mg of protein in 24 hours to allow participation in the study * Participants must have an International Normalized Ratio (INR) =\< 1.5 seconds above the institutional upper limit of normal (IULN) (unless receiving anticoagulation therapy) documented within 28 days to sub-study randomization. Participants must have a partial thromboplastin time (PTT) =\< 5 seconds above the 'institutional upper limit of normal (IULN) (unless receiving anticoagulation therapy) documented within 28 days prior to sub-study randomization * Participants with known human immunodeficiency virus (HIV) infection must be on effective anti-retroviral therapy at randomization and have undetectable viral load within 6 months prior to sub-study randomization * Participants must have asymptomatic serum amylase =\< 2 x ULN and serum lipase =\< ULN obtained within 28 days prior to sub-study randomization. Asymptomatic is defined as having no signs and/ or symptoms suggesting pancreatitis or pancreatic injury (e.g. elevated P. amylase, abnormal imaging findings of pancreas, etc.) * Participants must have ad