I've been there 4 times for a total of 57 days. I've rented a golf cart for 1/2 a day in all that time. I walk almost everywhere and have always stayed in town and last time was actually at the Mayan. (you'll be happy with it)
Now, I do have some regrets because every time I've been, I meant to take a large tour up and down island. So far, I just haven't made it. Too Belized out I guess.
My recommendation would be to rent the cart for a day or two only. Unless, you plan on going thither and yon every day, there is no reason to have it all the time.
I wouldn't even rent a bike for the whole time, but then I'm not a family that might enjoy that. Don't forget you have to keep a pretty careful eye on your bike.