We consider a finite-size topological Josephson junction formed at the edge of a two-dimensional topological insulator in proximity to conventional superconductors and study the impact of Cooper pair symmetries on the electron transport. We find that, due to the finite junction size, electron transport is highly tunable by the superconducting phase difference φ across the junction. At zero frequency and φ=π, the setup exhibits vanishing local Andreev reflection and perfect normal transmission due to the interplay of finite junction size and formation of topological Andreev bound states in the middle of the junction. We reveal that this striking behavior enables odd-frequency Cooper pairs to become the only type of pairing inside the topological junction that contribute to transport. Our paper thus offers a highly tunable detection scheme for odd-frequency Cooper pairs.