Well, in fairness, he doesn't call Cass for comfort or support. Note that Babs frames her concern for him in the context of Cass being unsettled; the reason Bruce comes looking for her is to comfort her, to make it clear that he wasn't being weird around her because of her, but rather because of his own nonsense.
He didn't intend to admit any 'weakness,' ask any reassurance, or in any way permit himself to rely on anyone for anything when he approached Cass. He went after her for her sake, not his, and had Cass not been involved, I strongly doubt he'd have turned to anybody, no matter who called him.
(Which kind of makes me wonder how deliberate Barbara's involvement of Cass in the conversation actually was. She's clever enough and knows Bruce well enough to have done that on purpose, but I don't know if she's necessarily self-aware enough, since that's the kind of thing you'd have to pull on her too, so it may not be a trick she's aware of just because if she were she'd be more aware of that pattern in herself than I think she was at this point).