I have heard good things about Sambuca, which is a few blocks north, I think.  I've never called, nor been.  I intended to take DS to the city this spring and would have called prior to that trip, but girlie's rather early arrival put the kibosh on that outing.

We have slightly different allergies, though, so YMMV.  It would be worth a call, I think.

if I remember, there is a Chipolte near Times Square...

There is.  It's on 8th Ave south of 52nd, but that's 20 blocks from the museum.  I don't know of anyplace to eat around the museum.  It's in a predominantly residential area.  Then again, it's been 7 years since I've take DD there.

I really don't know of any specific places near any of those locations.  Believe it or not, as of the last time I was there, there isn't really a lot of kid-friendly eating within easy walking distance of the Museum of Natural History (though I admit it's been several years since I've been there). 

There are tons of eateries on Madison Avenue around the corner from Mt. Sinai.  DS likes a deli near there - I have no idea of the name or whether it would work with your allergens.

There is a Shake Shack near Times Square - check out their website for allergen info.  There's a Hard Rock there as well.  There's tons of pizza all over the city, don't know of specific ones to recommend. 

Good luck!

DS1 goes to Sinai and Sicherer in two weeks. To soften the blow DH is taking him for a dad date to the Museum of Natural History while in NYC. Looking for a pizza place within walking distance of MoNH safe mainly for PA, would love TN and egg safe, too. Egg cross contam probably okay.