Are you sure that rolling on his back was submission? This can be a fighting position for cats, even though it never is for dogs. When cats are submissive, they usually crouch, not roll over.
There is something often called "petting-induced aggression," which is actually dominance aggression. That might be it, although I've never heard of a cat that redirects aggression to a dog when petted. I've never read on what to do with a cat who attacks dogs due to dominance, only cats who attack humans or other cats, and I'm no expert on feline behavior. You might want to look up cat dominance anyway, as something might help with the dog, but most of it relies on asserting the victim's alpha status. (Irony: a behaviorist realized that cats can be dominant when he accidentally gave a client a form used for dogs, and the client filled it out showing classic canine dominance signs...)
Here's the Dumb Friends League's advice on introducing cats to other animals: http://www.ddfl.org/behavior/catintro.htm
Reintroduction may work with enough counterconditioning, but it could be time to consult a GOOD behaviorist. They can also prescribe medications to make reintroduction, for example, go more smoothly.
Hopefully someone who knows more can give better advice.