I would talk to you ped first as mentioned previously. Isn't there some sort of stuff you can put on their nails to prevent biting that makes them taste bad? It think that's your safest option.
My SIL owns a salon and there are some things by law a reputable salon cannot do (such as use permanent hair color, peroxide and I assume acrylic nails) to small children. The chemicals are extremely harsh and could be dangerous to your child. When formulating these products, they assume the customer will be an adult and at least 85 pounds. Even then, some people will argue that they are plenty toxic.
I had fake nails done years ago, and I cannot imagine a small child would sit through the process willingly. It takes a long time to do and is rather unpleasant. The nail beds are filed to roughen them so they will take the glue and in doing so the cuticles are often roughed up. The nails are then glued on and then clipped. They use crazy glue to do this so you know it's pretty harsh and stings the eyes. Then they put on the acrylic stuff which, again, stings the eyes, smells awful and is not so good for you. Once this is done they then file the nails to shape the acrylic they put on using a dremel tool. It's lound, could be scary to a child and the dust from the acrylic goes everywhere. Nail techs wear masks since it gets into the lungs and can cause problems. I think this would be a bad idea for any child and if a salon said they would do it, I would personally run and quickly out their door.