Dogs do what works and if snapping at the cats works then she will do it more.
Best to take things in hand right away, so let's teach her that the food is not hers to protect and the cats are as much of the family as she is - they can share.
I would practice the leave it drill with the dog - start with a low value items and work your way up to a higher value items. When she shows that she understands (which should take minutes not hours or days) then it is time to introduce the cats. Be sure that she is listening to your words and you are not relying on the leash or a device to impress her. Then she is ready.
Make sure the dog is on the leash so you can stop any indication of attitude. Have the dog further from the bowl than the cats are and use 'leave it' if you need to remind her. Remember that catching the thought before it becomes an action is very important. Much easier to stop a dog who is just thinking about bad behavior then to stop them when they are already acting out.
Let the cats approach the food (could be canned cat food to lure them), and if she gets upset remind her to 'leave it' and use the leash to stop her if you have to. Do not hold her back with the leash because then you are doing the work and she is not learning to hold herself back. The leash simply stops her and then goes loose again. If she gets crazed then back her out of the room, remind her about the ‘leave it’ and reenter the room. When we correct a dog for making a bad choice we have to go right back to the situation and help them learn how to make a better choice – otherwise you end on failure and that doesn’t help anyone. You might have to repeat it a few times but always return to loose leash so she can think about what she is doing. You might choose to stomp a foot at the same time you say 'leave it' to help empower your word.
She needs to get the message that all things belong to you and YOU choose who to share them with. Food is not hers to protect. As she gets the message you can move her closer to the food as the cats eat. Your leash is short but loose so you can still stop her if she forgets her manners.
When she can handle being 2 feet from the cats at her bowl then move on to the next step.
Take the food bowl with you to the couch. Invite the cats to come along – you might want a helper for this. Have the cats on one side of you and the dog (on a short loose leash) on the other side. While the dog is in a ‘sit position’ offer her a taste of the food from your hand. If she is gentle and good then praise her warmly. If not then help her calm down before she is permitted to take any food. Then ask her to ‘stay’ and direct you attention to the cats. Offer them a taste of food. While you are doing this you need to keep subtle focus on the dog. If she shows any sign of becoming agitated then you give her a reminder to ‘leave it’. The leash is there as an emergency brake if you need it. If she advances to the cats you could also stand up quickly and walk quickly towards her backing her out of your space. This is your dinner party and rude guests don’t get to stay. You can calmly invite her back and start over. She needs to learn that her assertive behavior doesn’t work and calmness does. Eventually you should be able to share food with the cats, parrots and the dogs like we do every morning.
*Note - the white spot in my hair is a downy cockatoo feather not poop!
Love Them & Lead Them,
~Elizabeth & Doug
Dog Training the Way Nature Intended