The filter looks like an 'activated' bridged-T. The leg of the filter with the caps is bootstrapped from the output of the op-amp via the boost pot and its stop resistor. When SW2 is engaged the amp goes from unity gain to a gain of 2, so it wouldn't be hard to create a +ve feedback loop that is causing oscillation.
When gain is set above unity to create a resonant peak, it's not usually as high as 2 (well, not in my limited experience). Obviously someone has got it to work with that gain, though! I think a lot would depend on layout. It could also be that the wires to your boost pot are too long and are picking up stray capacitance (shielded wire would help here), but more likely that your problem is too much gain in this stage. Try reducing R8 (the 10k resistor from output to - input). You'll probably find that oscillation stops somewhere in the 2k2 - 4k7 range, but see what you can get away with (make the resistor as high as possible). Alternatively, make the stop resistor (R6) bigger. This might be the better option if you want a little more kick to drive the inverters.
As far as the boost pot having no effect goes, perhaps the frequency boost is in the wrong range for the pickups you use? Play around with cap values.