No, it wasn't going to be awkward to get all... girly carried on their date. No it was totally cool. Absolutely. Maybe they could just like... hold hands? Didn't Superman and Batman hold hands? Wasn't that how it worked?
No, Kara probably knew better than anyone how it worked and if she said that was how it needed to be, then he was going to trust her. She'd trusted him when they went on their first date after all (breezing through the Speed Force could be a scary thing to some people).