Things, especially but not only living things, have value beyond the price at which they can be bought and sold - always assuming they can be bought and sold. The value of a specific pet cat is not the cost of a generic kitten.
The purpose of damages is to restore you, as far as money can, to the position you were in before the wrong-doers actions occurred. Your duty to mitigate is to do all that is reasonable to minimize the costs - that may mean that you have a duty to choose treatment plan A over treatment plan B if A is cheaper than B and the prognosis is similar - it does not mean you have to choose euthanasia.
The defendant is free to argue that you acted unreasonably but given that vet bills typically run to 20x or more the price of a new animal, I don't think they would be successful.
Of course, if the animal were an economic asset, like a farm animal, rather than a pet, then a purely economic outlet may be reasonable.