May not be able to install a MaxxAir since it attaches to the frame of the vent, and if the frame is pushed in as OP states the MaxxAir cover may not fit.
What you might be able to do is go to the home center and pick up some
FlexSeal tape and tape the cracks in the cover closed so it doesn't leak and then deal with it when you get home.
So frustrating when you specifically tell someone to watch out for things like this and they happen anyway. If the shop can fix the start issue, tape up the vent, enjoy your trip and have the vent replaced later.