I haven't been able to fix it yet, but I did finally figure out what the issue was. If I turn an outer layer while the inner layer isn't completely aligned, (Like doing U after an Rw) something pushes it out of place... and it only needs to be misaligned by a 1/3 of a cubie. After that, the center piece isn't there to hold it back anymore and it can pop out.
Here's a picture I caught of a pop during OLL Parity. The pop occured while doing a U2 after doing Rw'
View attachment 3606
I don't think it's the edge piece that catches it. It might be the base of the corner piece. But it could also simply be friction between the small piece and the big internal piece... dragging it along during a U2. I only really started understanding the issue last night. I'm going to play around with different tensions and lubing. If excess friction is the cause, that should be the remedy. Short of that, internal catching is the only other explanation... that and sucking at 4x4.