It was probably going to be awkward, but Bart didn't mind. He was pretty much awkward all the time so this wasn't really a change for him, though he didn't want to seem uncool around Kara. At least not too uncool.
"Okay, but if I'm Batman, you have to be Batgirl." He said definitively- although he knew that wasn't really how it worked.
He nodded and lifted his arms so she could grab under them (thankful that he'd put on a LOT of deoderant) and nodded, "Okay! Let's go!" He tried to forget that he had this slight, little, tiny, itty-bitty fear of heights.