I think a key word in your husband's comment is "commit to going out", and anyone with little kids knows that's hard to commit to anything. Either they can't get a sitter, the sitter cancels, or someone gets the stomach bug.
So it's POSSIBLE that your SIL was thinking that your daughter as sitter would guarantee that the date would occur. It's hard to know if your SIL expected your daughter to a) sit for free or b) even have you cover the dinner (She said, "What a treat" so I hope she's not implying that you should treat them to the meal or the sitting - but then again, you have to be sure that what was said didn't amount to an invitation where the one doing the inviting does the paying.
I think you are right to question whether you are overthinking this whole thing - but I understand it's hard to figure it out when you have a difficult relationship with them and everyone is trying either to NOT offend the other or to look for reasons to continue to be offended. But since you are saying that you hate the growing distance and it seems to be your initiative (with your husband) to see more closeness, you may have to make more of the first move.
I think you should sit down with your daughter and establish some ground rules about what you expect and don't expect from her in terms of family harmony. If her donating her time or you paying her would establish a precedent, it's best not to go down that road. It's also fine to say one of the simple sentences suggested below, along the lines of "I'll ask her and find out her rate" but that puts you in the middle. Best to say, "It's probably easiest for you to make your own babysitting arrangements with her or anyone else so you can discuss timing and rates" or "I don't handle Susie's calendar - she's 16 and makes her own schedule." Maybe they assume you will drive the sitter right to them and bring her home, but that's only if the dinner date is in their neighborhood. What if it's near you or halfway in between?
So I'd stay out of it re the babysitting, and be careful how you word your invitation. "Would you like to meet us at XYZ Restaurant? I read and review and am anxious to try it." If they have a better suggestion, fine. You can alternate on who chooses the restaurant. If she asks about sitting, pause and breathe. You don't have to answer right away. It's okay to be surprised and to wait a moment, then say, "Gee I have no idea what Susie's plans are for that night. She's always so busy with her friends and her babysitting jobs, I can't keep track." Then say nothing more. If, somehow, Susie gets roped into babysitting and if they don't pay her, then she's absolutely not available the next time. "Sorry, I have plans for that night. Hope you can find a sitter."
Good luck - let us know how it works out.