What is cause my pantry LEDs to turn on slowly, and how can I mitigate it?
The power supply probably takes a little while to wind up.

If you can, break into the inline switch of the LEDS then:

Completely disconnect the door jamb switch from your current outlet wiring. The outlet should be rewired so it is always on.

Break into the LED wiring and run the wires from the door jamb switch to connect in parallel to the inline-switch.

Switching the DC is A LOT SAFER than switching the mains anyway.