I would accept the apology, make a friend, refuse the money, and the next time, if it happens again, when you call and say, "we've got an issue," that will be the tell.
Truth: Everybody's dogs get out sometimes. Not a perfect situation, but it's just the truth.
Loose dogs do not make good neighbors. Dogs are the worst chicken predators around here.
We have a right to shoot the dogs. Would we do it the first time? No. Would we do it the second time? Not if the person was actually trying to keep them in, and if I could stop them another way. But I would let the person know, if they come back, we will shoot the dogs. There is a neighbor up the road who takes no responsibility. The whole neighborhood is gunning for her dogs. One paw on my property (or the neighbor's at the back--I have permission there too)--and those dogs are toast.
Not sure where you are, but here, there are "neighborly" ways to begin the relationship. Around here, the possibilities would fall into two groups: 1-take the money, have less of a friend, no hunting on his land. 2-Don't take the money, make a friend, hunt his land.....#2 makes for better long-term neighbors.
Sometimes extending grace is the better way, is more diplomatic, and ends with better results.
If it happens again, and you take a harder stance, you are not seen as an unreasonable neighbor if you give grace the first time. HOWEVER....IT MIGHT HAPPEN AGAIN.
If it happens over and over, that is not a good neighbor, and all bets are off.
Every situation, every region, rural to suburbs to city, is different,
so you have to decide based on the dynamics of all this where YOU are. Not what anybody else thinks.
ETA: I should say, There is always an excuse:
"They burst out when I opened the door" --The dogs need to be trained to not do that. That costs money and/or time.
"They got through the fence"--You need a stronger fence. That costs money AND time.
"They normally come back as soon as I call." --Normally doesn't count if it's not always.
"I thought the gate was latched."--Double checking is easy--if you want your dogs to stay alive. It's about money and time and...accountability. Some people just don't take it. Others do.
The list of excuses is endless. An irresponsible person feels bad in the moment, but not enough to actually keep the dogs under control. Sometimes it goes well for a while, and then they slack. It can get ugly.
Just some experiences we have had.
It won't take long to figure out whether the neighbor is all talk or takes his responsibilities seriously.