I don't think you can rent a vehicle in the U.S. and cross the border traveling on to Belize. There is a definate problem with insurance to the company ....most would never take the risk.
Been there....done the drive....won't do it again!
My recommendation is to have your stuff trucked down to Belize either by Sterling or a private trucker that does this regularly....(I have the contact info)....and you two fly down. Meet your stuff when it gets here.
You will save yourself a lot of headaches this way!