I updated ltspice and ran the simulations again, this time I added a third model with a 1K gate resistor. The first interesting thing is that this time around, the current is not limited to just 2 to 3A when using the mosfet

it's actually more than the single 50m\$\Omega\$ resistor (similar value to RDSon of the fet). adding a 1K gate resistor seems to limit the current slightly, but not by much. a 10K one will reduce it to a peak of about 170A but I'm not sure if that's safe for operation. reducing the capacitor size helps but that also reduces the output current capability of the charge pump at the end.

I don't want to build the circuit, only to have it blow a dozen fets because I wasn't careful with the design, but here is the circuit that I'm planning to build (with a -12V LDO post regulator which I didn't include here):

I once tried it with TO-92 BJTs (2N2222 and BC-327 40). a single phase design only had a 0.185W capability (12V at 15mA. I'm sure it was constant power since putting a 500ohm load, dropped the output voltage to about 9.6V). I tried adding two more phases to it, but it just kind of died, I'm not sure why. so I want to use higher current devices such as fets in the next version, but I'm not sure if they will die as well or not.
Edit: the push pull amplifier design is needed since the DC40106B is only capable of driving 1 or 2mA at 12V.