At risk of this appearing twice. I had to make a small edit and the Forum is misbehaving at the momenet, so my comment disappeared, but this is a copy. It may or may not work.
Did you try contacting some of the other rescues who weren’t quite so close. (I take it that you rang the BHPS and got a list of contact details from them?) Some will come and collect hogs, or have volunteers who they can call on to collect for them. Some hog carers/rehabilitators know more about hog problems than some vets do, so it’s worth making a few phone calls if you haven’t already.
Re. the vets. I am always hearing that vets aren’t supposed to charge for wild animals, but I have also heard of people who have been charged for hedgehogs, so it sounds as if it is chance whether they do or don’t charge. It probably also depends on the vet whether they just check the hog over and give it back to you. Some may have carers, who they have connections with.
Just one thought. Make sure that the hog has enough room that it can get away from the hot water bottle should it wish to.
Caring is not really my area of knowledge, but, if you do end up having to look after the hog, and having had it checked for internal parasites, etc. (you might want to take a poo sample with you to the vet so they can check it). i.e. if it has a clean bill of health, I would continue to do what you have been doing, but without a hot water bottle. You will need to be prepared to clean it out every day. I wouldn’t want to release it now until it has put on some weight. (the recommended weight for release for a hog who has been in captivity for a period of time is 600g) If you do need to keep the hog hopefully someone with experience in rehabilitating hogs will be able to give you some further advice. I imagine your local carers will be able to advise you even if they can’t take the hog in.
Good luck. Let us know how you get on.