How to reduce audio noise from Bluetooth module
It sounds as if the bluetooth model is indeed injecting HF spikes back UP the power supply which then gets into the amplifier. This seems to be strongly backed up by the fact that things are OK when you use separate power sources.
(If the amp is class D, it may well be that the problem is the other way - the switching noise from the amp gets into the bluetooth receiver. Or some combination of the two. In any case the solution is the same.)
The problem is that you won't be able to decouple this away because the series impedance along the power rail between the two modules is close to zero. So your power supply filtering has nothing to work against.
The best solution, other than using separate sources, will be to use LC filtering. Now the inductors create a high impedance to the noise, which in combination with caps, will decrease the interference between the two modules.



simulate this circuit &ndash; Schematic created using CircuitLab
DCA and DCB are the power rails going to the two modules. The values of the C's and L's need to be determined by considering the frequency of the interference and also (very important) the DC current of the two modules, so that the L's do not saturate.
You might also have to be careful with the layout of the DC distribution. Where you have long traces or wires, you don't want HF current there, because it will act like a transformer to any nearby circuits. So you want the LC filtering near the modules, with the caps large enough to supply all the transient current needed by the modules while maintaining stable enough voltage. This is typically a bit of a trial and error exercise. Sometimes more than one stage of filtering gets you there more easily.
Side dish : it's just down to the way that different circuits draw power. Class D amps are basically switch mode power supplies that change output voltage very fast (simplistic explanation but OK), so these guys draw current in a very &quot;spiky&quot; way with a lot of HF components.