Don't quote me on this....but if hiring a vehicle using your credit card, it usually means you have free insurance for the vehicle you are driving.
(hire companies usually forget to mention that when selling you their insurance package).

You might want to check your credit card details. ie can depend on whether silver/gold plat/ plat + etc etc. As carts are the rental vehicle and as gas powered are considered vehicles you might be able to use that with your Credit card company...if so the hiring company is gonna be SOL wanting $12k.

I'd also be looking at whether you signed a waiver or was even given the chance to buy insurance. (I have no idea what the rules are in Belize for vehicle hire btw)

Hope your CC will cover for you or court find the vendor over the top.