I did find a work-around, in case anyone else finds this thread with the same problem.
Since the transitions I am using act like filters and affect the entire clip (not like the built-in Adobe transitions), the piece of the first clip that has the transition needs to be cut off the main clip and placed over the second clip, like this:
V2: ——————–[Clip 1b]———–
V1: –[[[[[[[[[[Clip 1a]]]]]]]]]]-[[[[[[[[[[Clip 2]]]]]]]]]]–
That’s what was causing the issue. The second clip on V2 (Clip 2) was scaling incorrectly. In order to fix this, I removed the effect from Clip 1b, nested both Clip 1b and Clip 2, and then added the transition effect to the Clip 1b nest. This seemed to trick Premiere, and it now displays the clips properly during the transition.