It came across as odd IMO, like the others were actively ignoring her. There were a few times where she'd have a line to the group about something - and no one responds. Add to that Sarah refers to her as Bat-Lady, and there's a sense of not exactly hostility, but definitely avoidance. I doubt that was the intention, but that's how it came across to me.
Really, he should have been one of the Paragons IMO, since he's a show lead - but whats done is done I guess.
Paragons:
Supergirl - show lead
Flash - show lead
Batwoman - show lead
White Canary - team leader of ensemble, basically show lead
Martian Manhunter - support character, Supergirl
Lex Luthor - villain, Supergirl/Superman
Ryan Choi - audience stand-in
I can see why Earth 1 Superman missed-out, despite getting his own show soon, two supers is a bit much and like what they did with Lex Luthor. But Black Lightning was the only other show lead to miss-out.