I don't have an answer for you on the strictness of the policy, but if I were you I'd hedge my bets. You can get an expedited passport in 2 weeks for only an extra $35 (plus overnight delivery fee). That seems like a small price to pay to avoid potentially ruining (or at least complicating) your vacation in Belize!
Here is the web site for the US State Department Passport Office: http://travel.state.gov/passport_services.html
There are answers there to all kinds of questions, plus you can download the renewal form.