I tend to endorse what David has said here. To me, the simple act of getting Aaron's birth certificate (£7) from GRO is a "must" to lay to rest that this birth is the right one and to get the true age.
I have no knowledge/comment on the USA bit. All I can say is that census records confirm that Aaron is "missing" from the 1851 census, despite the rest of the family being around. Aaron would have been in the order of 11 years old by then, and obvious possibilities are (a) death, (b) he is lodging with friends/relatives and there is a transcription error that 'hides' this fact on the 1851 census, or (c) emigration.
Possibility (a) could be true, although the only burial record is a long weay away in London (unlikely but possible). Possibility (b) has around 20% chance [I have found that transcription errors for 'Stock' are in the order of 20% measured from the number of corrections I have reported to Ancestry]. Possibility (c) seems to be provable by the recommendation David makes about USA documentation. I cannot think of obvious circumstances under which such a young lad would go to USA without parents/siblings etc. But stranger things have happened and it must remain a clear possibility.