I tackled the plumbing project today, after all. Yeech.
It took awhile to loosen the bolts (and working in a cramped space under the bathroom sink didn’t help much), but when I finally got the faucet apart, I was able to determine pretty quickly that something in the central valve assembly was messed up. So, we ended up with a new faucet for the bathroom. Same brand (Moen), but with a slightly different look than the rest of the bathroom faucets.
Putting it all back together was relatively easy compared to getting the old faucet out. What a royal pain in the ass it was, though, overall. Not a process I’d look forward to repeating anytime soon.