Non-surgical Nose Job on a Child for Very Low Nose Bridge?

It certainly is possible to inject fillers in the nose of a 12 year old. And this procedure would likely not interfer with the growth of the nose.
Still, I would be very hesitant to do this and it would require VERY compelling reasons to go ahead with the procedure at such an early age.

The short answer is it can be done. There was a time when any nasal surgery on children was thought to be a bad idea. Mainly because of concerns about adversely affecting normal facial development. However with time, the general consensus has shifted and now its not thought to be malpractice to perform nasal surgery on preteens. Much of this attitude change should be accredited to Dr Crysdale from Toronto. So injecting a temporary tissue filler can be done without undue risk to the child. 

However should an injectable rhinoplasty be done is another question that needs to be addressed. The question surgeons would have to the parents is 'why are you considering this?'. At 12 years old, the parents are still responsible for making these medical decisions, with the input of the child. Once this question is satisfactorily answered, it will be clear whether it proceed or wait.

There have been multiple disastrous complications associated with injectable fillers placed in the nose and is not advisable to have fillers injected into the nose at this time. It would be best to have a formal rhinoplasty performed when your son is the appropriate age of 16 years.