This paper proposes a comprehensive framework designed for the autonomous inspection of complex environments, with a specific focus on multi-tiered settings such as distillation column trays. Leveraging quadruped robots equipped with roller arms, and through the use of onboard perception, we integrate essential motion components including: locomotion, safe and dynamic transitions between trays, and intermediate motions that bridge a variety of motion primitives. Given the slippery and confined nature of column trays, it is critical to ensure safety of the robot during inspection, therefore we employ a safety filter and footstep re-planning based upon control barrier function representations of the environment. Our framework integrates all system components into a state machine encoding the developed safety-critical planning and control elements to guarantee safety-critical autonomy, enabling autonomous and safe navigation and inspection of distillation columns. Experimental validation in an environment, consisting of industrial-grade chemical distillation trays, highlights the effectiveness of our multi-layered architecture.