you could wire into the overhead lights, but as you mention, the lights need to be on for the fan. Here is a likely alternative, worked in my outback 295RE.
There may or may not be an unswitched wire running nearby. If not here is what you can likely do:
if you pull the microwave, you'll likely find that the wiring to the stove fan and lights drops down from the ceiling behind the microwave to the stove fan, or comes up from down below and then goes to the stove fan. So now you have a always on 12V source.
If needed drill a 1" dia hole or so into the ceiling behind the microwave. Then fish a wire from the vent over to the microwave. Now you have an unswitched power source for the vent.
Myself and several others have done this on outbacks.
If