Ld, OP is asking what the judge in family court can do, about the dog, at the request of his ex. I don't believe Family Court will touch any part of this.
OP is not concerned about the other courts. I would not be too concerned either. His ex cannot file anything regarding the dog. The boyfriend can try, but if I am not mistaken, OP's homeowner insurance will come into play on that.
On the basis that there is an agreement for the ex had to provide 24 hour notice, the boyfriend essentially was "breaking and entering". I don't think he will win any lawsuit against OP based on the facts given here. The insurance company can review the video and will be able to figure out how to use it.
I think OP should consider pressing charges against the boyfriend for breaking and entering. How is that going to look on boyfriend's military record? And I wonder if there is a way to play that video for his Commanding Officer.
I'd certainly be checking all those things out.