Thanks for the suggestions, guys! If I have to solder on the passenger's side, I'll give the shrink wrap + liquid wrap approach a try.
Also, "SHINK WARP" is awesome. Sounds like the sort of problem that can only be solved by reversing the polarity on your trilithium crystals.
Now that the driver's side 12v wiring is more or less squared away, I've been working on the insulation and framing the door & window. Shaving the insulation down to the height of the studs (it was something like 1/8" too thick <eyeroll>) is the sort of repetitive, fiddly task that is halfway between "annoying" and "meditative." And those darn foam shavings have the worst static cling I've ever seen--they blew everywhere and then
adhered, just out of sheer cussedness. Anyway, we're almost all levelish now. I've got a few small pieces left, and then I'll fill in the gaps with puffy foam and THEN it will be ready for the inner skin to be cut and installed.
I finished installing the framing, too, though I'll need to route the studs down to match the contours of the door and window. I'm planning to use my lovely bottom-bearing router bit for this task, hopefully tomorrow.
Oh, and I almost forgot! I was at the Texas State Agency Surplus warehouse the other day, when I stumbled on this lovely orange vinyl chair. For NINE DOLLARS. Yes, one of its seams is splitting, and yes, it's clearly about 50 years old, and yes it is exceedingly, even criminally, orange. But still. NINE DOLLARS. And the casters work. So I've now got an official Shop Chair, so I can do my staring, frowning, muttering, frowning, staring, and knuckling my forehead from the comfort of an upholstered rolly chair.
Artisanally hand-shaved insulated. Well, shaved by a hand holding an oscillating tool:

The Throne:
QUESTION: When I'm sticking the inner skin to the studs+insulation, what's the best adhesive? I was planning on using PL Premium, though I've also got some MasterSeal NP-1 on hand, if that's preferable. I'm looking for something thick enough to compensate for any irregularities in the insulation and strong enough to hold the skin tightly in place so that it won't give or make a crunchy noise if we lean against the wall. I want it to feel SOLID, and I'm planning to more or less paint the entire inside of skin with the stuff. Any thoughts?