I saw a lot of dog whisperer and Cesar will get the dog lying down calm submissive state. Then he brings the other animal but facing away from the dog and he lets the dog sniff the butt. Then still keeping the dog in a submissive state, I would put a bowl of dog food next to you and the dog and put the cat down to sniff the food and then let the cat take its time and leave. If you can do all that and keep the dog on its back relaxed then it should get better.
Most of the time I think its dogs and dogs but I saw him do it with a bunny and a dog who would eat small animals and it worked.
You can even go as far as the dog ignoring the cat while it sniffed or ate from the dog food bowl.
For the cat food room, I would set a 'invisible wall' boundary further than the door so you are able to go in without the dog. Invisible wall can be done by standing tall and face your dog but no eye contact or talking. and maybe hold out your hand to the side as if to motion blocking and wait for the dog to sit down. If he runs away, there may be bigger issues, you'd have to walk after him. Eventually he'll stop because there's not much place to go and repeat the above steps of standing tall facing the dog. All this done without saying anything or physical contact or fast movements.
Hope this helps.