I have purchased a new 1.2 ROM and have it in the unit. It responds the same way as the 1.0 ROM.
I have been able to repeat the results: If I turn it off, battery out then on start I can briefly adjust all parameters. I set it to the master tune, adjusted it up +4 and left it for several minutes, just to see what would happen.
The K5 maintained flashing the +4 until I moved the data dial and we are back to limited selection. When in that state, I can rename patches, change patch volume, but I cannot save with or without battery in, with or without memory protect on/off. I also cannot change System parameters.
Briefly, everything is as expected, then limited control. I was bale to set MIDI channel, then back to no luck. I have to leave it off for about an hour to get it to briefly allow System changes.
I think perhaps one of the RAM chips is failing or somehow being shut off, this could be as you mentioned a power supply issue. I have schematics from Kawai, there are several RAM chips on both the M-03 and M-04 board, but I think It could be the working RAM.
I'm also wondering if the RAM is getting written too by the system and filling up.
It doesn't help that I have an early revision, with jumpers soldered here and there and oddly the ROM in the alternate slot. I purchased another K5m motherboard and it has numerous differences, would not work with the K5 shell I have I think because of the power headers in different locations.