I think Ollie would actually be pretty chill about her, provided Babs was acting in a baseline-working-with-other-heroes way. She's a creepy fascist with no respect for the sacred right of privacy, and she's every bit as curt and uninformative as Bruce, so he'd rant about her and tell her off plenty, but he does that with a lot of his buddies. Bruce, in fact, is one of the people he respects most, and he probably calls Hal a Nazi as much as every other person in his life combined, and that's his best friend. And a well-written Ollie would care a great deal about how much Babs means to Dinah.
Babs, though, would not behave around him. She'd be snippy and obnoxious and provoke him as much as possible, and he would let her. Because Babs, honestly, is often not very professional about working with people she doesn't like. Her behavior toward Kory and Shiva, for example. Bruce may be a dick, but he's family; Ollie, on the other hand, has hurt family in the past, and Babs was the one who had to pick up the pieces. The friendliest she's ever been about him from the start of BIRDS on out was to respect how important he is to Dinah; she's never seemed more than tolerant of him for himself, and is frequently disparaging. No way that wouldn't come out, messy, if she were regularly in a position to give him orders.