Silver Age, or not, I cannot see Bruce letting Bette be a part of their team without some amount of his mentoring. (but, in your favor, Betty often just showed up and was a secret/surprise - no one knew she was doing it perhaps. So that added to her eventual 'stalker-ish' element.
I can't agree. IMO, he has no more authority over Betty than Kathy has over Dick. And besides, didn't he and Superman do a partner switchup (Batman working with Jimmy and Superman with Robin) - Jimmy wasn't trained. And if Batman respects Batwoman's abilities, then he should think her training of Kathy is sufficient.