Compromise is possible.
"When he is here with the dog he stays in his room and leaves me to sort the dog out, if I call him him to sort the dog he does but it would be left to me if he didn't."
So when he's home, he slips back into "son mode" where mummy does everything? Not criticising, I've slipped back into daughter mode myself once or twice, but I do think it should be recognised that this is a thing. He's 18, so he's only got two modes so far - son mode and military mode.
It's time for him to gain a new mode of being - independent adult mode. This means he does not retire to his room (games console?) and leave everything to you. He actively cares for his dog, because that is the adult thing to do. He walks it, he feeds it, he grooms it to reduce moulting, he trains it to stay off the furniture, he takes it into his room with him. He does not leave anything to anyone else, he does it all himself. That is his part of the compromise.
Your part of the compromise is dealing with being "slightly allergic I had sore throat and had runny nose". I'm probably more tolerant of this than most people as I spend much of the year like this. I would try other allergy tablets, for example I find loratadine more effective than citirizene, DH is the other way around.
"He told me that it was fine but he wouldn't be able to come and see me anymore if he couldn't bring the dog."
And what else is he going to do? Where else has he to go? Live on the base 365/year? As per @PuttingDownRoots's explanation, when he is on leave the dog cannot stay on the base it has to go with him. Where else will have him and his dog? It's a clumsy attempt at emotional blackmail befitting an 18 year old. Again, he has to learn independent adult mode. He has to provide accommodation for his dog. For leave of 2 weeks plus, the compromise I've outlined might suit. For a weeken/week, he needs to scope out the kennels local toyour home and his base, and find one he likes. As you said, "for a couple of weekends a year I think he could put her in kennels. He earns a good wage so money isn't the issue."
You bringing your allergy under control is the only accommodation you need to make to him. For everything else, he needs to step up and be an adult. He took on the dog, not you, so the dog is 100% his responsibility.