Lunch near the New Museum Contemporary Art
Since tomorrow is Saturday, both Freemans and Schiller's will be serving brunch, not lunch.
The no fish thing shouldn't be an issue. But I think you might have a short wait to get seated at Freemans dependent upon what time you arrive.
Schiller's takes reservations and is on OpenTable; I see a bunch of free tables right now.
The New Museum is on the Bowery right by the edges of Nolita and the East Village so you have options there as well.
DBGB, Hecho en Dumbo, Saxon and Parole aren't very far away, and neither are Public, Babaloosta, etc. What kind of cuisine are you looking for?
Freeman Alley, New York, NY 10002
Schiller's Liquor Bar
131 Rivington St, New York, NY 10002
210 Elizabeth Street, New York, NY 10012
299 Bowery, New York, NY 10003
214 Mulberry St, New York, NY 10012
Hecho en Dumbo
354 Bowery, New York, NY 10012
Saxon + Parole
316 Bowery, New York, NY 10012