It seems to me your only realistic choice is to fly from the International Airport to San Pedro. Your should be able to make that flight. After you figure in hotel costs it will be about a wash compared to other options.
There are now two private water taxis to the Island. (as of March 1st)
Tropic Ferry http://tropicferry.com/
and a new one Nautilus Water Taxi http://www.nautiluswatertaxi.com/
Both will pick you up at the Airport and deliver you to the dock where you are staying. (Assuming it is on the water) If not, they will get you in the general area. The trip is 1 to 1 1/2 hrs. after you are underway in the boat. So at best you will get in just after sun down and if everything doesn't work out just right you will be traveling in the dark which is doable but the above concerns are correct.
I don't think your $100 price range is realistic though. Don't forget that gasoline is much more expensive in Belize than the USA. If you decide to stay over night on the mainland I stayed one night several years ago at Global Village which is near the airport. I think they will pick you up for free and it was safe and OK. Last week I stayed at the Ramada Princess and they run a shuttle that will cost you $15bz each from the airport. Nice enough and very safe. The Ramada would also put you not too far from the water taxi the next morning.