Kara crinkled her nose, "tell Batman? Seriously?" that was a big fat NOPE, "First off he doesn't trust me as far as he can throw me, which might be a little far, but you know what I mean."
She remembered when she was found, Clark was the one excited for family and Batman insisted she could be evil. Rude.
"Secondly, that's creepy," she replied, regarding Batman knowing more about her body. Also Batman wasn't exactly the fatherly advice type. As far as Kara was concerned he just grunted a lot and told you how to train. Granted, he did help set her up in Metropolis, so it was a possibility some feelings got through that cowl of his.