I think I need to change my logic when talking with my wife about paying for dog-related damage
Why is there an argument about the dog related damage?
I wanted to have my dog neutered, but she didn't agree.
If the damage is caused because the dog is unneutered, as she insisted, then she should be the one taking responsibility for the damage. The dog would be much safer and probably better behaved if she agreed to have him fixed.
What if I am not the one to blame and it's her fault?
That's what I was saying, though it's a bit hard to put it into words.  I think that the damage caused by the dog might have been due to the dog's rowdy nature, and that having had him fixed would have prevented this damage.  If that is the case, then your wife should take responsibility, as she essentially allowed it to happen.