So you must think your kids did do it, if you are paying up, right? IE you know they had them out and were playing with them?
I was in a similar situation......
I have a BB gun and used to play with it......in my back yard.
Well, a long while after I put the thing away (what with having kids I dont like having it accessible and I really dont need it), my neighbor comes and asks me if I still have a BB gun. I said I did...he tells me that someone shots some BBs into his pool....right thru the side. It was a side not facing my place, and he is on the side of my house, not in back. Guess he thought I did it? I said "sorry to hear that" and moved on. I knew it wasnt me.
Anyways, he replaced his pool under insurance....(still not sure why he couldnt have just repaired the liner, but thats another issue)...turns out he finds out later it was a neighbor kid up the street who was pissed that he fenced off his yard and so the local kids couldnt cut through it anymore.
That solved THAT mystery.
How large was the window?
Unless you knew for sure that your kids did it or were playing with that BB gun recently, I wouldnt have offered to pay.
That all said, you already did ...but $565 to replace a pane of glass is ridiculous and he is taking advantage of you unless its a HUGE pane of glass.
ANY pane of glass can be replaced by a pro....the window type is irrelevant.
What I would bet happened was it was an old POS window to begin with....he called a glass place and they said "We can replace the glass but it will cost $250 and would be better to replace the whole unit since its old and crappy and we cannot guarantee our work on an old crappy window. A new window unit would only be $500...." He figures "Well its his kids' fault I have to even have it done so he should eat the cost of the whole deal." which we all know isn't the way it works.
Ask to see the estimate for having the pane replaced and THAT is what you should pay. Better yet, take the old window yourself to a glass place and ask for an estimate, then you will know.
As far as paying it to avoid a nightmare...well, this kind of neighbor will be an issue going forward once he knows you will submit.
Better hope he doesn't get a rock chip in his car's glass or something....he'll be at your doorstep again saying "Hey, you still cutting your grass with a mower? I think you kicked up a stone....."