Definitly keep the money outside. The Belizian dollar is worthless ouztside of Belize. Pegged 2:1 to USD. A black market exists that trades cash against USD and MXN at lower rates.Belize was good enough for John McAfee......until it wasn't.
Like many of those places, even if you plan to stay there long term, keep most of your banking and assets outside, just in case.
You never know which neighbors you get, and what connections they have.
The local banks do not have USD accounts. There are some "offshore banks" licensed to carry USD accounts within the country. These are also to avoid. See prominent insolvencies of Choice Bank and Atlantic Bank. If I would have to pick one I would go with Ashcroft's Belize Bank International.