Logic locking has received significant traction as a one-stop solution to thwart attacks at an untrusted foundry, test facility, and end-user. Compound locking schemes were proposed that integrate a low corruption and a high corruption locking technique to circumvent both tailored SAT-based and structural-analysis-based attacks. In this paper, we propose Fa-SAT, a generic attack framework that builds on the existing, open-source SAT tool to attack compound locking techniques. We consider the recently proposed bilateral logic encryption (BLE ) and Anti-SAT  coupled with random logic locking  as case studies to showcase the efficacy of our proposed approach. Since the SAT-based attack alone cannot break these defenses, we integrate a fault-injection-based process into the SAT attack framework to successfully expose the logic added for locking and obfuscation. Our attack can circumvent these schemes' security guarantees with a 100% success across multiple trials of designs from diverse benchmark suites (ISCAS-85, MCNC, and ITC-99) synthesized with industry-standard tools for different key-sizes. Finally, we make our attack framework (as a web-interface) and associated benchmarks available to the research community.