The battery is removed, and My MP-1 is powering up fine and retaining settings.

The picture is worth a thousand words. (Or, in this particular case, 8192 words of 8 bits each):

ADA MP-1 Batteryless Mod

Now I just have to just re-rack this thing, restore my presets which I saved over MIDI, and resume shredding.

Top left, you can see the desoldered battery, removed and lazily leaning against a capacitor, having nothing to do any more.

Bottom right is the daughterboard holding the new skinny chip, developed according to a careful interpretation of the chip's data sheet, in consideration of the ADA MP-1 schematics, and the data sheet of the original TC5565 chip.