How do you have your setup wired? How about a schematic? (hand drawn is fine)
Can you please share the schematic and layout of the board that your 328p is on as well?
The Uno and a barebones ATMega328p should be identical for this purpose, assuming you did everything the same way - so something is different.
. I am concerned that you say you were supplying 3.7-4.2 volts to a bluetooth module that is spec'ed as requiring 3.3 - that's probably too high for it, but I can't seem to find a real datasheet