Just thought I’d update this since I seem to have fixed it. (Fingers crossed.)
I bought a relay socket and appropriate color wires and terminals to bypass the relay in the TIPM in favor of an outboard relay. When it came time to trace the wires, it turned out that the colors in the wiring diagram were wrong. So I was forced to use my multimeter to probe the connections at the PCM and TIPM and determine which wires were the correct ones.
I cut the signal and output wires at the TIPM and routed them to my new relay socket, verifying continuity to the TCM connector. Then I made a new, fused connection to battery positive, and a new connection to battery negative, and connected those to the new relay socket. After verifying continuity to those, I put a new relay in my new relay socket, replaced the battery, crossed my fingers, and turned the key. It wouldn’t start until I did a hard reset of the computer, but after that it was fine.
I went on a 30-minute drive to get the engine up to temperature, then shut it off, waited a minute, and turned it back on. The symptoms did not return. The transmission shifts normally and the MIL does not illuminate. I made another test drive, about 20 minutes to refuel and back, and the symptoms have also not returned.
Strangely, my new relay’s Ground wire rings to both Ground and +12 with the key off. This worried me but maybe Chrysler uses some kind of floating ground system.
So it looks like I managed to hack my way out of this problem, at least temporarily. If I want to make it permanent, I’ll have to find somewhere to mount the relay (it’s currently just zip-tied to the body near the PCM) and route the wires (+12 runs between the engine and the air box, Ground runs between the body and the air box). I left plenty of length on the new relay socket wires to mount the new outboard relay wherever I want to; for now the extra lengths are just coiled up, zip-tied together, and sitting on top of the TIPM behind the air box.
Anyway, I hope this information is useful to anyone who has the same problem and is tired of replacing TIPMs. New TIPMs aren’t available anymore, and as my experience has shown remanufactured ones are an expensive roll of the dice.