Bring as much cash as you exchange rate.

There is a limit to the amount you can withdraw each time from an ATM. I believe it is about US$250.

There is a limit to the amount of money you can bring into Belize without declaring it. I believe it is US$5,000.

Capital One is the best card to use outside of the USA. Remember to call your credit card provider and tell them you'll be using the card outside of the US. Learned this the hard way...they refused to authorize the card after the first use.