Grumpy_Mike:
If that is an N channel FET then it will not work.
Yo need a P type FET and base resistors and to put the LED and resistor in the collector of the transistor and not the emitter.However when you have done all that you will still have trouble because the PWM signals will beat with each other, some work at diffrent frequencies so they will not add up like you hope they will.
Hmm. So, what would be a better solution? Hook each color up to a PWM output and tackle the software code?