Could you please post some other information (through hole or SMD components? layout or at least a schematic)? It would be much more easy to find a solution, there are lots of experienced people here (not me).
Although you may already have done, I'd suggest putting small value caps between supply and gnd pins of all the ICs, with very short leads/tracks. If the voltage dip is the cause, a big capacitor on the supply of the bluetooth module could help, also together with a small resistance or inductance in series to split the two supplies (I'd put it on the side which has less current transients, so I suppose on the PIC side):
heavy cap || bluetooth module
supply /
\
inductor/ferrite bead/1-10 Ohm resistance -> heavy cap || (PIC&friends, each || with small cap)