You are pretty much stuck on Maui for the night. I suggest you take a flight on Mokulele Airlines the following morning, they have one that departs Maui around 7:40. Small planes, no TSA and are used to hunters going to Lanai.
Website HereThere is a Marriott near the airport but pricey so Maui Beach Hotel is another option that has shuttles.
Maui Beach Hotel Don't really expect a beach.
I live on the Big Island but work on a normally uninhabited island south of Maui (Kaho`olawe) and I pretty much fly Mokulele and if I have an over nighter it is at Maui Beach Hotel. You can have dinner at the hotel or walk over to the mall across the street. The Mexican place is pretty good.
You do know the deer on Lanai are small, right? Not much bigger than a German Shepherd.
Hope that helps.