Good thing they've already agreed that it wasn't your fault. As heartless as this sounds, if they try the whole 'but you just killed our dog' crap to try and make you feel bad to avoid paying you for the damage their pet caused, you can always throw back 'You could have kept your pet secured and avoided this whole mess to begin with'.
I'm not saying it's their fault, but if they try and make it seem like its -your- fault, then let them have it.
People can be completely unreasonable when it comes to money. Everything could go well, but if it doesn't, then mention to them that if you can't come to an agreement, then getting the insurance company to deal with it will actually cost them more. Should work.
One thing though, after whats happened.. keep an eye out for any dodgy stuff happening to your property.. Neighbours can make life a real pain in the bum.