I would NOT let FIL babysit....quite simple....i will always errr on the side of caution no matter what. Offer to let them spend time at your house....and watch them carefully....abusers are good...many don't give off a "vibe".
As for your DP...he needs counselling....for himself as much as anyone else.
My DH was abused as a child...and we have talked about a lot of things. H once said he did not think he loved DS...so we got into a big conversation about what it is to love a child....he has always heard about that "new love" that you have with a newborn (sort of like the romantic love honeymoon period) and thought if he did not have that...then he did not love. Then we started talking about all the things they do, the things they look forward too, how he would feel to lose him...and we quickly realised that he did love DS.
It took us several goes to find the right counsellor...but the hunt was worth it.