UNO + unstable PWM + Linear Led Drive

Very strange diagram, these are hard to follow so you should post a schematic.
Why do you have reed switch in the ground?
If the external input is 0 - 10V you are only feeding it with a 10V PWM signal at best, not an analogue voltage. What is the FET part number!

You need an RC filter to make the PWM into a smooth DC signal.

the capacitance in the transistor is working against me?

Again meaningless I am afraid.