We investigate the topological quantum compilation of two-qubit operations within a system of Fibonacci anyons. Our primary goal is to generate gates that are approximately leakage-free and equivalent to the controlled-NOT (CNOT) gate up to single-qubit operations. These gates belong to the local equivalence class [CNOT]. Additionally, we explore which local equivalence classes of two-qubit operations can be naturally generated by braiding Fibonacci anyons. We discovered that most of the generated classes are located near the edges of the Weyl chamber representation of two-qubit gates, specifically between the local equivalence classes of the identity [1] and [CNOT], and between those of the double-controlled-NOT [DCNOT] and [SWAP]. Furthermore, we found a numerically exact implementation of a local equivalent of the SWAP gate using a sequence of only nine elements from the Fibonacci braiding gate set.