I do not believe that it is illegal to possess a spear gun in Belize, since all the dive/snorkel/ boat operators have them and use them for the wonderful picnics.
Spear guns are not "firearms" within the meaning of any gun restrictions anywhere to the best of my knowledge.
No countries, to the best of my knowledge, confiscate spear guns on sailing and other vessels.
However...I can't tell you what the customs officers at the international airport would or would not do if you brought one in that way. The best way to find out for sure would be to contact a customs broker. I think there is one advertising on this web site, if I'm not mistaken.
Perhaps someone on this board has the newest version of the Belize customs regs handy. And of course, customs officers are often unfamiliar with their own regulations.
Perhaps Gaz or someone who has a boat business and presumably buys spear guns will know the answer to this question.
I know that we brought in a couple of the spear holders (not the guns themselves) for a friend a few years ago and had no trouble with bringing them in, nor were they opened.
However, I wonder why it would be necessary to go through the hassle of shlepping a spear gun to Belize. If you want to go fishing for lunch, the guides on your boat will be equpped with them and they will be happy to let your husband try his luck.
You will of course have to check it as there's no way you will get it through security at any US or internatinal airport, and they may charge you extra. Why not leave it at home?