For starters I would like to commend your dog on having a soft mouth or good bite inhibiton. Next, your dog is acting like a normal dog, this in no way reflects that the dog was abused
far from it! this is a form of survival for a dog. The dog that eats lives to see another day. It's a natural instinct to want to hord and eat what you can when you can as you never know what tomorrow is going to bring. Your dog is just doing what comes natural. The secret is, teaching your dog to do what doesn't come natural. Do not put tempation in the dogs reach and "expect" that the dog should know better...that is putting human emotions and rational thoughts into your dogs mind. You dog doesn't feel bad about guarding food, the dog does not realize it is doing something wrong. The dog is reacting to your body language after the fact...in no way is the dog putting two and two together and thinking...hey, my new owner just smacked me because I was eating the cats food again...I better rethink my day to day living now. All that you are doing is reinforcing to your dog that YOU are unpredictable and attack without warning. Your dog bit you, but, you have also attacked him...wrong move, you attacked your dog and in your dogs mind it was unprovoked...you turned on your dog. Please, don't put human thoughts and emotions into your dogs head...you have a dog, not a human in a fur suit.
You have received some good advice. resource guarding responds well to desensitization and counterconditioning. I really urge you to pick up a book called "Mine! A practical Guide ot resource guarding in dogs" by Jean Donaldson. You need some professional reading material, or seek the advise of a professional animal behaviorist who specializes in positive training methods. This is not something to take lightly especially since you have children.
For the time being, don't have any valuable resources or cat food where your new dog can get it...it sounds as though you are already going to do this. Don't leave potential hazards around that may cause the dog to guard and possibly attack again until you have professional help...the next time it may be one of your children. Impliment the NIFIL policy...but, don't count on this as a solution to your problem. You need to learn to countercondition this problem with professional help.
Last, don't expect more from your dog than a human...and just because you have a resource guarder does not mean the dog was EVER abused. I have a resource guarder here at my house. I spent a year desenitizing him, he came from a great breeder in a great home where food was plenty...he was never abused, hurt or ever had food withheld from him....it is just his natural instinct to protect his dinner from the other dogs....I now have him so he will eat side by side with all 4 of my dogs...5 dogs all from different backgrounds and they all had some issues...they eat side by side daily with no issues at all. I had to work at it, and I used the NIFIL policy as well as some other techniques such as while he was eating having one of my other dogs sit beside me while I dropped great peices of meat into his bowl. He soon learned that good things happen when the other dogs are around. I can even give each dog a RMB with no issues and they will go out into the compound and chew side by side. It takes time and patience to teach a dog to go against it's natural instincts.