Turns out that I had two problems:
1.) Not priming made the acrylic not stick to the plastic as well as it can
2.) Mr Masking Sol Neo is really harsh on acrylics.
As a test, I had some parts that were still painted WITHOUT primer. I tried masking them with Tamiya tape instead of Mr Masking Sol Neo, and everything went fine.
For the parts that I had to strip down and repaint, I primed them with Mr Surfacer 1200 and the colors on top of the primer are now MUCH more durable than they were on top of just plastic. Additionally, Tamiya tape's straight edges give a much cleaner line, no matter how neatly I applied Mr Masking sol.
Tamiya Tape + Priming = success.
Thanks again to everyone for the suggestions and help!