Some suggestions:
1. Switch the bed back to bang-bang control, using the B1 parameter in the M307 command.
2. Try connecting the LED strip directly to your power supply instead of to VIN. That avoids the resistance of the leads from the PSU to the Duet causing voltage drop at the Duet when the bed heater turns on.
3. Use a better-regulated power supply (probably in conjunction with (2) above).
4. If your 12V PSU has sufficient extra power capacity. turn up its voltage to 14V, then use a 12V low dropout regulator to provide a better-regulated 12V supply for the LEDs.
5. Use a separate PSU for the LEDs, or a separate PSU for the bed heater.
6. In a future firmware revision I may make the bed heater PWM configurable, then you will be able to turn it up to 70Hz or greater, high enough for the eye not to see.