Prevent MOSFET input being low when connected to output of powered off device
Its just as you said, you can use an N fet connected to the diode resistors and to ground, that has 4 advantages:

1: Your problem gets instantly solved, and

2: N channel MOSFETs tend to have lower resistance than a P channel ones of the same size

3: You don't need a pull up resistor which consumes power when the MOSFET is off (not that it is a lot of energy tough).

4: if you decide to use many diodes in series hence increasing the needed led PSU voltage the circuit remains unchanged (unless the Vds of the chosen fet is too low).