Hello Anouk.... I don't have a lot of experience in the Student Visa area.... However this is the website to immigration.....
http://ins.gov.bz/index.php/visa-requirements/general .....Even though I believe in your situation your child might be able to attend one of the private schools here on a simple tourist visa..... I would check in with this group as well; they are a popular grade school with other expat children attending.... to include our French friends....
http://belizeeducationcenter.com/contact/