Update & some history:
Back in June or July, when I had the topper put on, they installed a 12V outlet fixture, which is wired into the taillight, so it's always on. It's not a lot of draw, but it's there, and they told me about it when I picked up the truck. Before my issues, I hadn't been driving much, cuz, well, everything was pretty much closed. After I posted my issues and got the truck running again, I drove it a bit and unplugged the DC outlet in the topper.
Long story short (too late), I haven't had the issue since. Remote start has worked perfectly, and no issues with stability control / stuck in first gear.
I'm thinking that when the battery gets low, in this case from parasitic draw, the electrical system goes haywire. It's ridiculous that a healthy-but-low-battery completely disables the vehicle, but, if you're having these issues, try charging the battery before losing your mind.