Nah. Catwoman only started being pushed as a love interest with the 60s. Before she was the villainess Batman would catch and put away. Even then Batman would have other love interests like Vicky Valley, Talia al Ghul, Nocturna and so on.
And Catwoman being the only one who understand and accept Batman is a load of bullshit. Catwoman's an amoral and selfish thief that steal and fight for the thrill of the danger and to suit herself. Sure, sometimes she's shown as a compassionate soul that care for poor people and only steal from black market art dealers, like a modern day Robin Hood, but that's not her usual character and only became a recent thing with Brubaker because he was trying to make her work as a heroine. Even then King's very run showed that Catwoman didn't even knew what Batman did on his day-to-day basis and wondered what he did for fun. So how the fuck she understand him? Other than the whole furry thing.
Nah, the Bat-Family: Alfred, Dick, Tim, Barbara, Cass, and so on. Even Jason and Damian. These people understand and accept Batman for who he is. Not Catwoman.
But i understand what King is trying to do. He has to sell this romance as the most important and big aspect of Batman's live, because everything rest on Bruce giving up the other most important aspect of his life for it. So he has to sell their connection as if it was almost life a life support for Bruce.