Great. Thanks for the additional information. I really appreciate it.
Unfortunately food aggression is one of the most common bad behaviors in dogs and is one of the hardest behaviors to break because the need to guard their prized possessions is deeply ingrained in them. Back when all dogs were wild, they learned to protect food and other resources as a matter of survival. Snapping at and growling at other pack members was their way of telling them to back off and show some respect. Even though Peter Falk is domesticated, this need to protect important to him things is still there.
The first thing to do is to get him enrolled in a basic obedience class. Not because you want him to learn to sit/stay (although that's great too), but because it will build the bond between you two and elevate you into the top spot in the pack hierarchy, because you'll be asking him to do something and then expecting that he follows through. It sounds to me like right now he's challenging your authority by stealing things that are yours and being aggressive when you try to get them back. By getting him into class, it'll show him you're in charge...no matter what he thinks.
Being a working dog, it's also very likely that he is testing your authority just like a human toddler would. He's trying to figure out if you'll yield to his threats. Sadly..if not nipped in the bud, this behavior can get much worse with the dog graduating to full on attacks to other members of the home. You seem to have a very good grip on how to keep him under control for other issues, so I have no doubt you'll be able to get this behavior stopped as well.
It's also important to mention that even though you may get frustrated, please don't yell at him or physically discipline him. This will make him even more aggressive because he'll feel like you're angry enough to fight for the food. Also, as much as you want to, please don't comfort him, sit with him, or baby him when he acts like this. That's actually positive reinforcement and will encouarge him to act out.
It's possible that this has been a problem for him for a long time and that for some time it was sort of hibernating, so it will likely take quite a while to break the habit. The first thing we need to do is establish a different routine for feeding. You (and any other human in the house) need to be the ones who are in charge. We're going to teach him that good things happen when he doesn't act aggressive around his food bowl.
If you're currently allowing him access to his bowl or any treats all day, please stop that. We want him to get used to eating at specific times, so removing his food bowl inbetween meals will be a signal that he needs to focus on eating when the bowl is on the floor instead of being aggressive. During this training period, you're going to be using two differnt bowls and we're going to want to feed him/train him in a low-traffic area of your home. Moving the bowl from room to room will also help keep him from becoming territorial of a specific room.
For the training, you're going to place the bowls out of his reach. Ask him to sit in front of you. When he's calm, place one of the bowls on the floor (empty) and then drop a piece of kibble into the empty bowl. Don't bend over to put it in the bowl..literally drop it. Do not drop another piece until he eats the one in the bowl. If he doesn't act aggressively, you can try putting a couple pieces of kibble in your hand and invite him to have them.
While you're feeding him, you're going to alternate methods of giving him the food..give him a piece out of your hand, and then drop one in the bowl. If he begins to get distracted, you can drop in a high value treat like a little chicken breast or slice of hot dog.
Once your dog shows no signs of territorial behavior around the bowl, you're ready for the next phase. Fill onw of the bowls with his kibble and place it on the floor. Call him into the room and request that he sit and stay. When he has been still for a few seconds, give him the "Okay!" command and allow him to have the food. When he's eating out of that bowl, take the second bowl and put a couple pieces of the high value treat in it..have it about 10 feet away from the other. Call him over to the second bowl. When he's eating, go back to the first and put more kibble in. Then back to the other bowl when he's eating that kibble. Keep switching back and forth until all his food is done for that meal.
You'll keep doing this training for several weeks...and over time, you can start moving those bowls closer together. Make sure you're paying attention to his body language and any verbal cues he gives you that he's being possessive. What we're doing here is teaching him that it's okay to give up the food he's eating because there may be something better around the corner.
I've followed these steps with several of the pitbulls I do rescue work with, and within a couple of weeks, their guarding behavior either disappeared completely or was drastically reduced. We turned mealtime into a fun game of doggy dining etiquette. They would happily leap into the 'sit' position, watch me put down the bowl, heed my 'wait' cue and my 'watch me' cue before approaching the bowl. Once I gave the 'okay' signal, I was eventually able to pet them as they ate, praising them. It worked for me and my most difficult cases, and it can work for you and your dog
If you don't feel that you can stop this behavior on your own, however, I urge you to seek help from a professional behaviorist. This is a serious behavior problem that can eventually threaten the safety of you, your family, your visitors and the other dogs in the home.
I hope this helps!!