I’m so sorry for your loss, OP.
I’d tell the SO to kick rocks but really, it isn’t worth your time and hassle. Just get out of the barn and don’t look back. Like @endlessclimb mentioned, if anyone asked your opinion on the barn and how it’s managed, you can share your experience. That in itself goes pretty far.
Unless the SO had any sort of ownership in the horse, they have no legal right to decisions regarding euthanasia outside of disposal of the body IF you planned on burying the horse on their property. If you planned to have the horse professionally removed, then they could have some say in when people entered the property to remove the horse. However, they have no legal right to your horse’s medical records and no legal right to make any sort of decision in treatment for your horse. They are overstepping their boundaries if that is what they are asserting.
Again, I’m very sorry for your loss and you should not be concerned about a shitty SO who is 850 miles away and making the situation worse for you. So unnecessary but the best thing to do is to cherish the good memories with your horse, find healing and move on to brighter horizons. Hugs.