Have had three issues where dog play got a little rough between my dogs and others and needed repair, once mine, twice others. The two that were others were in my invisible fenced backyard, one neighbor refused anything from me, claimed it was his fault (dogs knew each other and had played before, just got a little rough that day and he had just let his dog wander his yard), the other tried to soak me for all, I agreed to splitting the cost for neighborhood relations, even though the blood was in my yard and it torqued me he tried to claim my dog jumped the fence to attack his (it was winter, snow on the ground, easy).
For my dog, my ex invited a friend and his dog over, they weren't paying attention (another story) and my dog got his ass kicked, I did not accept any contribution from him as "they" were her guest.
From what I have read of this, I probably would ask the neighbor to keep his dog away due to the issues, but not ask for money.