Fanyo,
I’m sorry about this annoying journey. In finding out that this same issue happens with another pedal, I’m thinking that there is something fishy going on with your PXL. Normally the footswitch popping occurs when there is a DC offset on one side of a bypass capacitor. When the pedal is engaged the bypass cap allows some of the DC voltage change to momentarily flow through the cap, causing the pop. Pull down resistors can be used to help keep the charge from accumulating on one side of the cap.
If you just need a Boss style buffer, they are really easy to build. You can just follow the schematic from one of their service manuals (look at the TU-2 service manual, for a very simple schematic). You just need one FET and one BJT and a few other supporting passive components. You may consider trying to build one yourself, on some perf board. It would be very small and you could even install it inside the Quaverato chassis, so you could use the Quaverato’s 9V rail to power the circuit with. Perhaps a better solution would be to find out what kind of circuit the PXL is using and modify that. You could even install the buffer circuit in the PXL chassis so this problem doesn’t occur, no matter what tremolo pedal you use.
…I’m just throwing out ideas here. I hope your Quaverato ends up serving you well, despite this annoying issue.
Take care.
-Brach