How to eliminate Bluetooth noise in electric guitar
There are two ways to solve these types of problems. Either at the source and or hide the effect. Fortunately the whistling and clicking appears directly related to the RN42 and not the PWM's. 

Note the HiZ inputs of the pickup coils are going to pick up everything. From Florescent Lights to the RN42 aliased noises. Typically a band pass filter is used at the point of observation or A2D. Where here your pickups are likely directly tied to the AMPs input. And a filter here is undesirable.

The RN42 is creating noise that is bleeding out onto the power supply and radiating to Pickup's. And shielding the pickup's is impractical. Rather one needs to filter out the noise from its source. My own experience with the RN42 has a lot of clicking in the audio connected circuits. 

Recently Elektor's July &amp; August 2013 had an article for Android Elektorcardi♥scope where it used an RN42 to send the result to the Android. Where it was littered with Filters on the power supplies to all the IC's. Below is the filter used on the RN42. 







simulate this circuit &ndash; Schematic created using CircuitLab

Where the 30R@100MHz was used on other supply inputs with different capacitance. At DC it typically is under .2 Ohms, but with spikes and noises that will rise dramatically and react with the capacitance to block the noises.