Move your horse. The barn is convinced you are wrong. You are convinced the barn is wrong. It’s not going to work out even if it works out.
Unless you can take on full care for your horse and politely mend the rift, move to whatever back up barn you have in mind. Now would be a good time.
That probably all sounds awful and mean. It’s not meant to be. It’s wisdom gleaned from years of observation. It’s why I take full responsibility for my horse’s care spending far too much time on the road every day, 7 days a week. It’s also why I have walked away from lovely client horses on more than one occasion over the years.
Popping back to add that a puncture is not a cut or scrape. They have to heal from the inside out and if there’s anything nefarious left in any part of the tractor you want it to work its way out, not get trapped by surface healing. Keeping a wrap on until it is done draining is not wrong. It may not be necessary but it’s not wrong. You may not like it but it’s not wrong.
There may be an upcharge but I’d bet you agreed to that in your boarding agreement. Boarding barns don’t get rich from upcharges for doing annoying little jobs that take staff time away from other things.
A compromise until you get moved - turn out fully wrapped. If the horse is still swelling at night bandage overnight too. Change wraps every 12 - 24 hours.
Bandages do not push swelling upwards. They may slow the gravitational pill of swelling that is coming from higher up but they do not push swelling up.