That would be a small claims case. That means no lawyers are required. It also means that the judge might know little to nothing about horses. The judge might think they are like dogs or such and then find you responsible.
I personally do not see how it is your fault at all. If you want to pay half because you feel bad then do so. Otherwise I would call their bluff and see if they take you to court.
Before you pay anything be sure to ask for a copy of the bill so you are not paying for half of routine vet care items.