*Membership spots not really limited!
That's Willing Suspension of DISbelief.
And it requires things to be "believable and internally consistent". Your "Batwoman" running around on her own when there's a Bat-crossover going on breaks that believability and internal consistency. If she was REALLY a Bat-character, she'd be involved in that crossover.
That would be logical if DC actually cared about the character which I don't think they do apparently.