That will haunt you for the rest of your life.
I would also talk to an attorney. Better to go through insurance than your neighbor. Even great neighbors can become nightmares when money is involved. Trying to do a handshake agreement means absolutely nothing. And you don’t know what shots or anything yet. What if your wife’s dog wounds become infected and she has a 200k dollar hospital bill? You think your neighbors will cover that?
We have insurance for a reason. Call animal control, call an attorney.