Is this your first dog? You sound like you have very strong beliefs in one training method and are set in your ways but for some reason are asking for advice on a public forum.
Cesar Milan is great, but he's a PET trainer and the things you see on TV are dogs that have most of the time been ruined by owners that didn't know what they were doing. If you're doing some of his methods as preventative methods so that you don't get a "red zone" dog then that's fine, but the chances of you having that if you got your dog from a reputable breeder are probably very small.
At 3 months old, she's not challenging anyone. She's playing. A dog barking doesn't necessarily mean a challenge. It can mean a variety of different things. Most of the time its excitement or fear at that age. Trust us, she doesn't want to be the alpha at this point, and you don't need to go all Cesar Milan on her to establish yourself as the alpha. Just be a good, fair, confident leader and you'll become the alpha without having to destroy your dog's confidence.
I've spoken to so many people that believe their dog is an "alpha" when really its just that they don't have proper leadership so the dog decides to make its own decisions. They tell you how their dog is an "alpha" when it comes to other dogs and then their dog gets it handed to them by a more confident dog that just doesn't want to take their dog's crap. Really do some research on the whole "alpha" thing. Even Cesar doesn't use that method anymore, he actually never ever says "alpha."