Getting dog and cat used to each other without traumatizing my cat?

I've had my cat inside for about a year and a half. He started as feral and just showed up in the yard one day. When I brought him inside, I was watching a friend's cat who was on deployment. My dog was fine with my friend's cat and only chased her if she got the zoomies, and if he caught her he never tried to hurt her. Otherwise he'd just sniff her and go about his business. The thing is, that kitty never ran FROM my dog or tried to scratch him.

MY cat though, is afraid and runs from my dog, which causes my dog to chase him and then if my dog catches up to him, the cat scratches him which makes the dog want to fight. My cat has his own bedroom that's blocked with a baby gate, that has a smaller door at the bottom so kitty can come and go and the dog can't go in there.

I've tried getting my cat comfortable around my dog by putting my dog in his crate then carrying the cat around the house, but the cat just jumps down and runs back to his room. Would it be a good idea to put my cat in his crate and just have them both crated and in the same room together until they get used to each other? I don't want to traumatize my cat but I'm not sure what else to do.