I don't think you can get a permit until you are an approved legal resident of Belize. You certainly won't be able to have a gun "collection." Yes, it takes many months to obtain permit approval.

You can join the rifle and gun clubs which are all located on the main land if you want to fulfil you love of shooting.

There are very few safe spots to shot on the caye.

Probably you should keep you gun collection in the US and visit it once if while.

You could take up virtual/internet shooting!!