I don't imagine catching goats is that difficult - some variety if net would do. It's the transportation of your goats that might be tricky.

Transporting livestock absolutely requires documentation, and $17 is getting off easy. Bringing an animal to a new location where it could potentially come in contact with local wildlife and you better be damn sure of it's provenance, immunizations, testing, etc. It sounds like they're not requiring any of that, presumably because it's an isolated population and most of the requirements aren't needed, so all they need is the document showing where the animal came from. Of all the things to be mad at bureaucracy about this is not one of them.