This is by no means my area of law, and I don't even profess to know a lot about horses.
My understanding is that a lot of animals, including horses, can and do die during veterinary treatment where there is no lack of care or negligence. You would need to prove that your horse's death was due to a breach of the vet's duty of care - not just that the horse died because of their treatment. In other words, if the vet acted reasonably in the circumstances given the requisite degree of knowledge and understanding then it's unlikely that they're legally responsible for the death.
I would suggest that the first course of action would be to get an independent review of the matter by a qualified equestrian vet who can give you an objective report on what happened. This may, if possible, require an autopsy.