There are very good Japanese-inspired cocktails and snacks (miso wings!) at Bar Goto, which opened in 2017 to great acclaim. Try a post-date smooch after the sansho pepper cocktail :)
Attached to the Wythe Hotel, Le Crocodile is a French brasserie that will make the date seem fancy without seeming like a huge flex. You might accidentally make eye contact with an actor from Succession between bites of pâté.
Like the name implies, Tokyo Record Bar is Japanese (omakase and sake) and involves vinyl. This is an excellent one for a special occasion or some occasion-less novelty. Make a dinner reservation inside to get the full experience (dancing to Diana Ross on full-blast with four other couples you just met).
To continue the bath theme, check out this burlesque joint on a Tuesday night. It’s a low-lit, sexy vibe, and you can snap pics in their namesake copper bathtub if that’s your thing. They also have live jazz on Wednesdays if your relationship isn’t ready for gratuitous nudity.