You may need to lose the logic inversion of that circuit… If so a p-channel MOSFET can be used, driven from the existing open-drain output (and pull-up resistor). This then drives the main n-channel MOSFET (which has a pull-down). Also this configuration will be normally off when the controller is powered down - and doesn’t invert the sense of “on” from the original controller. Both MOSFETs can be non-logic level (which are easier to find).
Since you might want to switch quite fast and have a good driving capability of 2A, those pull-up and pull-down resistors can be quite low, say 120 ohm 3watt. That gives 100mA to the gates which will switch them off nice and quick.
So I’d suggest using the 12V supply for the p-channel source, 120 ohm from gate to source and gate driven directly by existing unit.
This then drives the main n-channel MOSFET (Rds(on) < 5mOhm, proper heatsink) which also has a 120ohm 3W resistor from gate to source.
If you get power resistors that mount onto the same heatsink this will help keep things cool.
Say load is 30A and 5mOhm n-chan MOSFET is used, then it dissipates a max of 4.5W, each resistor 1.2W, so heatsink needs to cope with 7W per channel (20W for R+G+B if 30A is the current).
Power dissipation depends on the square of current so you must do the calculations for your worse case current. Don’t be confused by the MOSFET current spec, this is not the relevant spec, Rds(on) is all-important.
If the PWM rate is high you will have to work out the switching losses too. Remember that during switching transients the MOSFET’s power dissipation spikes much higher than when off or on. Its important that such transients are much shorter in duration than the PWM period (a hundred times shorter in duration is a good place to aim for - this is why we need low value pull-up and pull-down resistors (MOSFET gates are VERY capacitive).