I tried your first suggestion, PRR, as it was pretty easy to swap a couple wires around. It was better, but still there. So I tried grounding out pins 5 or 6 on the IC, which disabled the trem. However, I'm using a SPST switching pot for the depth control, so I need it to do the opposite function and actually enable trem when I turn up the depth.
So I went the easy way and ran pin 4 of the IC (V-) through the switch to basically turn the IC on when I want trem. I don't know if it's totally kosher, but it works.