Moving back to this meter - I thought I'd install a line supply on the one which is working, get it out of the queue, and then use it to check out the one with the wonky range behavior, so I know which parts are good and which aren't. I'd be happy at this point to have 3 working 8600A units while I am doing some other work.
I tested the built-in transformer which is part of the battery-operated unit and no matter which taps I used, I can't get it to provide a rectified, filtered output of less than 20VDC. That's way too much for any regulator I want to use, and depending on the digital circuitry pulling the supply down to a reasonable value when loaded is not, to me, a good engineering solution. I figured that if I'm going to replace the transformer, I'll just make my own supply instead of faffing around with the weird Fluke topology. So I fished around and found a 12VCT transformer exactly the same size as the existing one. Set up with a standard FWCT secondary filter, it provides a solid 10VDC which is perfect for a standard 7805/LM317 regulator. I've tested it and it works, so the tough part was installing the transformer.
As luck would have it, the one I found has standard mounting ears but also a set of flanges which hold the outer frame onto the device. I bent the ears under so they'd keep the frame snug, and bent the little flanges down. They almost perfectly matched the ones which the Fluke-supplied transformer had, that allowed it to be soldered to the board and grounded to the mains neutral. (I'll show the process when I do the second 8600A; it happened so quickly I forgot to take pictures of the first one) If you look, you can just see the pins disappearing into the original board mounting holes at the corners of the transformer. So that was a fortuitous discovery and will allow me to fit everything in with no cutting or shaping. I'm going to put the small filter/regulator board where the secondary parts would go in a standard 8600A and just hook the output into the 5V rail where it would normally connect through the ferrite bead and front panel switch - so it'll work just like the original except no batteries to mess with.