The principle of using a single supply to emulate dual rail supplies is to have a resistor divider that gives half the supply voltage; eg. two 1K resistors in series. Add a capacitor from the mid point to ground, eg. 10uF or 100uF.
Then use the mid point as "ground" as far as the biasing of the opamp is concerned, so the + input would connect to that in your "typical connection" diagram.
The piezo discs are very high impedance, so the input resistor should be at least 47K.
Using an eg. 500K or 1M pot as the feedback resistor (output back to input) would give a gain from zero out to 10x or 20x gain out.
Add a 0.1uF between the opamp out and the Alesis input.
The Piezo triggers can easily give tens of volts though, so I don't know how linear it will be, with a 12V limit?
I'm surprised the Alesis unit has no adjustments? Mine (DM-10) has vast range of input tuning settings?
You may be better off trying a new piezo disc in the cymbal itself, and checking whatever touches the disc has not degraded or worn.
I have a virtually unused one I could check with a scope to see what the output voltage is, if I can remember where I stored it?
(I replaced it with the Pro X hihat on mine).