Your schematic should work for low PWM frequencies, your 'screaming', whatever that might be, is probably caused by a wiring error. Are you sure you connected the GND's of the 12V supply and the uC board together?

Other than that the circuit is suboptimal; it takes R3 a long time to charge the gate of M1 which increases dissipation. Replacing R1/R2/R3/Q1 with something like a TC4427 would be a better solution.

Regarding PWMming TEC's: you can do that as long as you keep the PWM frequency above 1kHz or so. If you PWM with a low frequency there is too much stress on the thermocouple joints due to them thermally expanding and contracting many times a second.

Wouldn't it be easier to just use a motor driver (module) which is essentially an h-bridge? Saves you the trouble of dealing with components.

@nxt-1 said in Idea: peltier hotend cooling?:

A slight advantage can be had due to the higher temperature differential between the ambient air and the hot side of the PEC

But the higher differential will only benefit you if the current heatsink temperature is somehow too high. And unless you add more fan cooling you won't be able achieve the increased differential in any case - so if the goal is to remove the fan then I'd be surprised if this was the way to go.

Not knowing much about the geometry, maybe heatpipes would allow you to transport the heat to an area where there is more room for heatsinks and/or fans?