Hire a babysitter so even you can enjoy dining out.

OP here. We had a 5:30 reservation at The Riggsby, and everything worked out really well! Everyone was happy with the food and cocktails, we had plenty of room for her car seat, and the restaurant was really quiet when we got there. We left a very full restaurant with smiles on our faces and a sleeping baby. Can't thank you enough for your suggestions!

For another Jose Andres restaurant, we went with my son at that age to Zaytinia. It was noisy enough (even at 5:30) that we didn't feel conspicuous at all, and with the small plates food came quickly and efficiently. It's also big enough of a restaurant that we were able to be in a corner with a lot of window space.

Le Dip will be fine - tons of kids in that neighborhood and restaurant is loud anyway.

I'm one of those diners that can get cranky when there are loud children at a fancy restaurant after 8pm. This is NOT you, OP. I think the fact that you are dining early and sound like an attentive parent means you should feel comfortable taking your child anywhere. You've gotten some great recommendations here.

I think PP above had a great idea in calling the restaurant to see if they have room to accommodate a stroller, or if they suggest you use a carseat during dinner instead.

Have a great dinner.

Dino's Grotto. We took our son to Dino's when it was in Cleveland Park starting at the age of about 5 months. They are lovely with kids. And not too shabby for a foodie, either!

You want a place that has a bit of noise. I would just let the BIL pick a few and select one of those. Or repost with his list of choices.
Just make sure if the baby cries, pick her up and take her for a walk, preferably outside, not around the restaurant, or find a corner. And don't breastfeed with exposed breasts. I'm not against breastfeeding but just do it discreetly. I've seen some women uncover and expose and then feed. Or ask first before you do it.