Well, my hunch was correct. My Playdate is unfixable. I'm pretty bummed about it, partially because it was my fault, but also because I'll no longer have the original Playdate that I preordered 10 minutes after it went on sale. Sigh.
Fortunately, there are a lot more pros than cons in this situation. I'll have a fresh, new Playdate; a fully functioning d-pad (the up button was a little mushy); and now I know never to do that again. I didn't even lose any of my save data, since I backed it up way before any of this even happened.
Hopefully I'll get the new one pretty soon, from what I've heard Panic's pretty good about sending out replacement Playdates quick. Pretty sure they say this to everyone, but they told me to send my Playdate back to them for inspection, which is really cool! Maybe they'll figure out what went wrong...
(P.S. Panic's customer support is the absolute BEST! They were super nice, and I definitely felt less stressed about everything when we were trying to figure out a solution to my issue)