Dog is obsessed with the cat but never attacks – just won’t leave her alone

We have a 3 year old cat and a 1 year old mixed breed 90 pound dog (possibly retrieved and pit bull?).

We did the recommended slow starts. Had them both is separate rooms. Then did the baby gate. Then did the baby gate with only the opening for the cat. We are still at this stage.

The dog shows absolutely no aggression to the cat whatsoever. However, he is obsessed with her. He will stick his head through the baby gate to watch her, run up to her to lick her, and follow her around. He will jump on counters to try and get up to her high places as well. He is always very exited, tail wagging, and just wants to be around her.

The cat, to be fair, doesn’t seem scared of him. Sometimes she sits behind the gate and just watches him stare at her, and she will be purring during this. Other times, if we have the doors closed, she will meow at the door to be let out to see the dog. The cat seems to enjoy following the dog around, but gets mad when he follows her, and will hiss and bat at him. She never uses her claws.

We are at a loss. The dog, if he sees the cat in his space or through the baby gate, is unable to focus on anything else. It’s obsessive. We can distract with food or toys but once they are gone, he is right back to it.

I am so unsure about what to do with this dynamic. It has been days of this and it is getting exhausting.

Any advice is greatly appreciated.