This could have been a redirected sort of aggression too. Nieko has been building up all this frustration (stress) towards something he can't get to (Pig which is also a stressor as a new addition to the house), then the added the stress of being yelled at, trying to be shooed away from what he is obsessing over and the fact that this dog is likely in some pain (disc injury? more stress) and you have a dog who is likely to go over his bite threshold. Your flipping him on his back just added more stress and to be honest, you are lucky he didn't bite you too. Your daughter is smart not to be willing to "punish" your dog like this.
This article can explain a bit more about bite thresholds:
"All dogs, whether they are defined by owners or behavior professionals as “reactive,” “aggressive,” and yes, even “friendly” can and will bite. A service dog or therapy dog can and will bite. The goofiest dog you’ve ever seen can and will bite. The dog that allows young children to climb all over him and pull his ears or tail with seeming aplomb can and will bite.
Bites are usually caused by an accumulation of stressors. Each time a dog is exposed to a stressor, stress hormones are dumped into the brain. These stress hormones are like the puzzle pieces in Tetris. They build up over time. You have to actively reduce the stress (like a Tetris player clearing lines) through management, desensitization, counter conditioning, and general stress reduction techniques. If you are not taking steps to reduce the stress, it begins to accumulate. The dumping of stress hormones into the brain leaves the dog increasingly sensitized to stressors, which replicates the puzzle pieces dropping faster and faster until you eventually reach the threshold. Soon, the dog bites. The game is over.
Stressors vary in individual dogs. One dog may be stressed by loud noises, nail trimming, men with beards, wearing a shock collar, foul weather, and a bad diet. Another dog may not seemingly respond to these factors but is sensitive to visits to the vet’s office, small children, cats, people that smell like beer, dogs walking past the fenced in yard, and people approaching or entering the home. Every dog has stressors (commonly called “triggers”) and a big part of effective behavioral modification strategies is identifying these as accurately and thoroughly as possible, which allows behavior consultants and handlers to focus their efforts most efficiently. Stressors, like Tetris pieces, accumulate over time."
How Are Dog Bites Like Tetris?
And these explain why you shouldn't be flipping your dog on his back to "show him who's boss":
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You do need to take action but not in the way that you seem to feel is appropriate (alpha rolling). I would suggest the Ruff Love program for Nieko and that everyone in the family participate in it:
Welcome to Dogwise.com
I would also suggest that you will probably need to rehome the pig. Your dog may have too much prey drive to happily live with a prey animal in the house. Also it does seem that the pig played a major roll in Nieko going over his bite threshold. Since the pig is just a baby rehoming should be easier. There are also rescues for pet pigs, just like their are for dogs. That might be your best bet for ensuring the pig finds a happy and appropriate home, since they are often "novelty pets" for people and are extremely difficult pets. This might help you find people in your area who can help you:
Pig Placement Network
