agree on the battery thing as first point of diagnosis.
yes you can plug the A3640 board into the A3k. they were a cheap upgrade for 030 machines for a time when A4000/040 owners were upgrading and selling off their A3640 boards. Plus it's not a bad idea to rule that out as a point of failure, but.
it is a bit of a slog as the A3k's CPU slot is under the drive tray/carrier - that is only easiest to remove after taking the PSU out.
also you'll have to play with some jumper settings on the A3k to switch clocks to the CPU board to get it to run off the installed CPU board rather than the internal 030.
another thing that might be worth doing if you can get one cheaply and easily enough is an A4000 to ATX psu adapter - rule out any power problems. check the ends of the power leads going into the mainboard that they aren't corroded or burnt.
pull out all fast ram. pull everything you can. get your machine to a basic working state first, then start working your way back up. My A4k resurrection process i recently went through to bring mine back to life - i ended up figuring out that all 4 of the fast ram sims i had installed on the board on it were dead or not-trustworthy. Somehow my DVDrom drive was toast aswell.
if your A4k is one with a simm slot for chipram, see if you can get some life with a different simm in there?
i wish you luck on your journey!