The 50Hz hum that's visible on the spectrum may be audible in some very low noise situations and it's coming from the mains power - if you choose to address it, it is a fairly common problem and some re positioning or changing of some of the initial power supply stuff should help. Given that it's the peak, even if it's not very audible, it's certainly a potential issue. Just physically moving input lines away from signal level stuff can help, as well as shielding any signal lines that go over mains lines, but stuff like bigger filter cap banks in the PSU or a toroidal transformer can help too.
As for the 3.2kHz tone, have you looked at a spectral analysis of that BT module current draw? Could be that the sharper edges of the pulses coincide with a 3.2kHz tone even though the base frequency does not, and since 3.2kHz is the 4th harmonic of 800Hz, then they are probably linked. There's also a chance that the 3.2kHz tone is a mixing product of much higher frequencies interacting.
Try isolating the digital power supply and ground plane. If you are working with a PCB, dividing the analog and digital ground planes can help reduce the digital related quick return currents from interfering with the more sensitive analog stuff. Otherwise, I'd probably put a small inductor in series with the bluetooth module's power supply. Something like main filter caps on the supply rail -> inductor -> secondary digital side filter caps, these LC filters can be a really good way of isolating noise from getting back out into a system or quieting down a noisier supply. Bigger inductors and capacitors mean a lower rolloff frequency, and if you were only filtering out MHz+ level noise, you could probably get away with a ferrite bead instead of a full on inductor. Of course, you want the inductor to have a low enough DC resistance and high enough current capability to supply your digital section, and you want the capacitors to be big enough relative to the inductor so that you don't get a big voltage spike when you first power up or shut down the device and small enough to prevent your supply from needing to feed a massive capacitive load (if that's an issue, like with switching supplies). A spice simulator can give you a good idea of ballpark performance for given values, but for dozens to hundreds of uF of caps on either side and just a few mA draw for the modules... maybe 10-22uH of inductance and less than 2 ohms of DC resistance? Just a ballpark guess.
If you look at some datasheets for low level analog designs (like with a sensitive DAC or ADC), there are some pretty good explanations of separating analog and digital ground planes and examples of decoupling power between them. They'll also probably show you some examples where using different linear regulators for each helps with noise isolation (so, if you have a 7.5V input or something and need a 5V rail, you have a regulator for the digital and a regulator for the analog). Then, there's also physical separation on the PCB layout, minimizing digital trace lengths, flood filling the ground plane, and even via stitching between digital and analog sections for that little extra bit of isolation.
It's a complex thing to really nail the execution, but there's a lot of sort of general procedures that can at least improve performance.