My main advice is you REALLY need to visit Belize before you can really make such a decision as move your entire life to Belize. Is it possible that on your husbands 6 week non working period you could spend a few weeks in Belize and A, You could find out if Belize feels right for your family. B, You could explore most of the major districts of this small country and possible check out some schools. C, You could talk to Other expat Families in various areas of Belize.
I believe there are a few South african expats in the country, if any are on this forum they may be able to give you more specific advice.
As south Africa and Belize are both commonwealth countries you should be able to enter and get the tourist 30 day visa stamps.
Not sure about the travel side for your Husband, as the majority of flights too and from Belize and the outside world go via the USA, I do not know if that would require your husband to have a transit visa for the US to accommodate ease of regular transit through. I know many years ago we had such transit visas when we lived here and had a lot of need for regular travel in and out of the country via the states. That all could be totally out dated info these days.
There are some good schools in Belize but you may be told other, as the curriculum is not very similar to that of North America which can lead to readjustment for children if they spend some time in Belize then have to transition back into that curriculum when the family return north.
I know of several families who ether totally home school or supplement the Belize schooling with additional on line school programs.Total home schooling is not the easy option, if you have no teaching experience or you are not a self confident organizer.
As to safety, that you would need to explore in the different the areas on the ground and to some extent decide on the atmosphere you felt most secure in different places. I have no personal fears where i live, and am often a woman alone in Belize for weeks at a time, also I live close to a vey good school, but I am a retiree and do not have young children so my experience is irrelavant in this case. Hope some of this may help.