Do you have any context info? For example, might the phone have got a bit wet recently or were you doing some particular task?
I don’t think that there’s any reason to focus on the bottom module specially at this stage? If it were a charging or loudspeaker problem then maybe, but from what you say it’s more likely to be the power button, main board, the screen or in between. Possibly some contacts just need cleaning.
If you wish to test an exchange of modules I don’t think either phone would be at risk, but in any case I think cleaning is the first thing to try.
Clean the contacts of the battery and of the battery compartment, with a cotton bud soaked in isopropanol or medical (90° high-grade) alcohol. Leave to dry 3h. Then try turning on again.
If no better (as is probable since you tested another battery), then
Disassemble all the modules.
Clean them all carefully with the brush attachment of a vacuum cleaner (careful they don’t get sucked in) or use a can of compressed air. Plastic or wooden toothpick for lint in corners (USB port, …)
Then clean all contacts that you can reach on the main board and on the modules with cotton bud + alcohol and then leave to dry over night. Reassemble and test.