Buddy is so cute! I love the look on his face. And yes, I too am one who manages to fall up stairs

As someone who has raised Chows for show, and adopted cats, then a dog, then more cats, they
will get used to each other and at least learn to live in the same household. I'm not saying they're gonna be best pals, though one of our cats is completely in love with our big dog (she has issues), but they will find a way to work it out.
When we first got Sushi (our Shih Tzu) 2.5 years ago, she used to chase the cats around, trying to get them to play with her since she was used to living with other animals in a barn (she was a rescue who had been in a puppy mill) and playing with everyone. Eventually, after many MANY whacks on the nose from the cats' paws (claws were
usually sheathed), she learned that they didn't like her and so she made do with us

We kind of have ownership of our neighbor's cat - long story short, they ignore him so he comes in our house

- he was absolutely FURIOUS with us for 2 months after we brought Alice (very friendly rescue from a feral spay/neuter program) home. He wouldn't come to us and if we tried to pet him, he'd hiss at us and stare at her. He attacked her a few times until he realized she's just a little softie.
They will work through it

I suggest sticking with it, unless there are major blood-drawing altercations, then something needs to change. We had a trial period with a young male that didn't work out because he hadn't been neutered long enough for our large male dog to consider him a non-threat to his territory. Then we had a female Yorkie who was just demented and did nothing but growl at anyone. She nipped my husband who was just trying to let her sniff his hand, so back she went.