Semen from other countries is imported into Thailand, now mainly from the USA. I used to know a Thai dairy farmer who use to go to Australia and bring back semen in a small 5-liter flask, himself, he could bring back a good few hundred straws, he then uses to sell them on to farms.
This was 10 years ago, I would say the rules and regs have not changed much.
You could as the Thai guy did, just go yourself, I do not know if you have any semen tanks, if not you can get them in Thailand a 3-liter tank would cost about 16000-baht, paperwork you seem to have covered, vaccines that would Foot and Mouth , now may be Lumpy Skin? which I think?? Australia does not have.
Importing embryos, not my field, but I would have thought that could be done yourself like semen, but the problem would be finding someone here to do the implants.
All the best.