Help with resistor in gain and filter op-amps circuit
The op-amp won't be driving the capacitor even if R4 were zero ohms. The other side of the cap isn't hard-connected to a strong voltage source at all - the input impedance is Gohms and the only other component is the 1k to ground. It'll work with R4 = 0.

If you choose your op-amp to have the right gain-bandwidth product then one op-amp can do the job but you have to bias the input (ditto your original circuit) with maybe a 1 Mohm resistor across the piezo. It comes down to analysing the effect of input bias currents through the 1 Mohm and deciding maybe that a different op-amp might produce less output voltage offset at a certain gain.

Or, you can still do this with a flakier op-amp if you insert capacitors in series with R2 (or R6) - this makes the DC gain of each (or the only) stage unity so DC offset amplification is minimized.