It doesn't sound like you're doing anything wrong. Without a battery, the unit will not remember the last settings and will just use defaults.
It's possible you might be facing separate problems for cold and warm, but I think more likely it could be the same root cause. I would try to tackle the booting issue first, and start with the processor circuitry.
As a prerequisite, besides the voltage on the supply rails which you already checked, take a look at each rail with a scope, or a DMM on AC volts. Make sure there isn't a significant amount of AC line ripple on any of them.
Then I would take a look at the /RESET line, on pin A3U30.19. After power up, it should go to TTL high almost immediately and stay there.
If all is good there, it might be worthwhile to spend some time with a can of freeze spray and try to pinpoint a component or board area that's sensitive to cold temperature when booting. You can use an inverted can of air duster if you don't have freeze spray.
If it's a mechanical problem (as coppercone2 hints), you could also gently (!) twist the A3 and/or A2 boards while booting to see if there's some effect on boot behavior. You could also poke at various board areas with a wooden stick. It could be a bad solder joint, or fine crack in a trace.
The service manual employs a signature analyzer for go/no-go checks on the processor and related circuitry. Do you have access to one? If you use Sigrok, it supports signature analysis, but to be honest I haven't tried it.
How about an EPROM programmer? It could be used to check the contents of the EPROM(s). EPROMs can experience bit rot.
And what serial number do you have? There are a huge number of design changes in these units listed in the back of the manual, all organized by serial number.
Also, if it's not too late, I'd recommend you mark all the adjustment pots with a fine Sharpie before moving them. These units can be a real bear to calibrate and it can be useful to get back to where you started.
EDIT: Small clarification.