Ah so I went back to not using Beat Snap just to double check, and the issue is actually different than what occurs with Beat Snap. I guess I got more familiar with the post-Beat Snap issue than the pre-Beat Snap problem.
What occurs when Beat Snap is off is that there isn't any looping, but the timing slowly drifts out of phase with the beat. If I turn 1/4 Beat Snap on the Composition it starts looping again, but less often than the clip level Beat Snap.
My final use case uses Ableton Link to get the beat from Rekordbox, but the issue persists even when Ableton Link is turned off. The problem is more obvious when there's music so here's a few examples.
Drift with beat snap turned off: https://vimeo.com/473267823
Loop with 1/4 Beat Snap on Composition level: https://vimeo.com/473268651
The loop happens at 1:02