Tidal disruption events (TDEs) have been found to show a preference for post-starburst (PS) and quiescent Balmer-strong (QBS) galaxies. This preference can be used to help find TDEs in transient surveys. But what other transients might "contaminate"such a search, and by how much? We examine all reported transients coincident with the centers of galaxies in the French & Zabludoff (2018) catalog of spectroscopically confirmed PS and QBS galaxies and photometrically identified PS and QBS galaxy candidates. We find that TDEs and Type Ia supernovae (SNe) are the only types of transients classified in the centers of these galaxies (aside from one active galactic nucleus flare), with Type Ia SNe being 8.3 ± 0.2 times more prevalent than TDEs (1σ confidence bounds). This factor is ∼2.7 times lower than in a control sample of quiescent galaxies. Narrowing the sample to spectroscopically confirmed QBS galaxies does not change these statistics much. In spectroscopically confirmed PS galaxies, however, TDEs are the ones that outnumber Type Ia SNe 2 ± 0.6 to 1. Unfortunately, there are few such galaxies in the catalog. By classifying transients from the entire catalog, three times more TDEs are expected to be found, but with a ∼16 times larger Type Ia SN contamination. We use the public ZTF photometric archive to search for possibly missed TDEs in the French & Zabludoff (2018) galaxies. We find three unclassified clear transients - none of which are likely missed TDEs based on their light-curve colors.