The flirting is necessary because if they're not flirting, then people will suspect as to why they're walking together if you're not making it blatantly obvious as to what you're doing. It's a tactic he's used multiple times in multiple situations. He can't help but feel a little disappointed that she doesn't know how to flirt back. The sarcasm is cute though.

"Forgive me for taking all of your luck then! I'll have to figure out a way to give it back to you somehow."

They're walking along and it's really starting to sink in that he wasted so much time being lost. God, he should have tried contacting someone, but there was no time! Getting the information in the first place had involved a lot more action than he'd liked, and then the desperate call to a contact of his to do some quick favors to protect his cousin had eaten up so much more of it, there was just... nothing else he could do.

"Glamorous and on a bit of a... time-sensitive schedule. Could you tell me what time it is?"

He knows it's not past the window, but he doesn't know how close it is to this information becoming useless.