It doesn't need to be a bass pickup. There's nothing inherently different between a guitar and a bass pickup except the number of coils and the coil spacing. Same magnets, same typical 42AWG wire. So a guitar pickup will pick up just as much bass end. Just use whatever's handy/spare/cheapest. Single coils will probably be better as they have less mid-range emphasis.

No need for a preamp or high impedance input then.

Most piezo-style kick blocks I believe are pretty solid as you really just want the 'thump' of the wood as your foot hits it. But a pickup style unit will need a thinnish top that can resonate - at least a bit. You don't want a long sustain and you don't want it so resonant that you get acoustic feedback when you apply a bit of volume.