Since you already paid the adoption fee for this dog, you are legally responisble for the adoption, but if you talk to the rescue organziation and tell them your circumstances have changed they might be able to pair you with a better dog for your family. You are doing a great thing, taking in a rescue dog. He can be corrected from jumping on the kids, but don't leave the kids unsupervisied around him, because he is a larger dog and doesn't realize that children are easily hurt. You have to teach him how to be around children.
This would be my question if I was the one adopting him to you. Are you emotionally and fiancially able to take this dog into your home, knowing he has these stomach problems, needing special food and meds, and that he is mouthy? I would also have you come out and meet him, with your children and god son, and your french bulldog to make sure they got along before I went through with the adoption. Did they do a homecheck? You might be getting more than you can handle if the rescue isn't on the up and up.
If you are unsure at all about this big boy, I wouldn't take him home. Bulldogs aren't easy dogs to deal with and tend to be dominate and french bulldogs are the same way. Are you talking an American Bulldog or an English Bulldog?
Personally, I think its great you considered a rescue dog, but I want to make sure you know what you are getting into before you bring him home, especially when there are children involved. A big dog like that could serious injure a child, just playing, not even meaning to. You need to make sure you can handle this dog with or without your husband at home before you make the choice to bring him home. I'm sure the rescue will understand if you can not bring him home because circumstances changed in your household.