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.

No way leave the kid alone.  I would not do any cosmetic procedure on a 12 year old, especially injectables.  The child needs to be evaluated by a craniofacial surgeon like the other plastic surgeon mentioned.
He may have some underlying pathology that you may not know about.

This is a very bad idea for several reasons, the most obvious of which is that you are proposing an elective, cosmetic procedure on a person too young to give informed consent. Rhinoplasties, surgical or not, are performed for RECONSTRUCTIVE surgery in children this age, but this is not the case based on what you are describing.

There are no guidelines regarding a minimum age requirement for using Injectable Fillers in your child's nose. The youngest child I've treated to date is a 16 year old, and his pictures are attached.
Feel free to email / post your child's photo and I'll be happy to give you my thoughts.
I hope this helps, and best regards.

No, not a good idea. The injection of any material can cause scarring. It can hamper the normal growth and/or make the rhinoplasty more difficult.
If you were to still proceed I would certainly stick with hyaluronic acid injections only, i.e., Juvederm and Restylane.
The bigger concern I have is that why the child is so conscious and upset about it. At 12 years of age if the child is so concerned about the nose it might point out to some underlying unaddressed pscyhological issues. If there is any chance this could be the case, please make sure you get the proper counselling for the child.
Regards