"Tag-along" stories like these are fairly common. Not trying to hurt anyone's feelings, but here's are a fews problems I have with these stories --

Food, clothing and arms.
Plus, stories like these are almost always first-person and post-war. Rarely if ever does one find a wartime letter or diary record or even a newspaper story about "Little Jimmie" or some other child who decided to tag along. It's always "Little Jimmie" who tells the tale many years after the war. Try to find a veteran's post-war memoir which mentions little boys in the army, children not even in their mid-teens. A wartime news writer would have pounced on such a story like a dog on a table scrap.

Wartime letters and diaries of the Army of Northern Virginia such as those used in "Lee's Miserables" frequently mention slaves brought along by officers as servants. Since rations and uniforms couldn't be issued anyone not carried on roll, officers took take care of servants out of their own allowances. When officer allowances were cut in 1864, not enough was left for a decent share for servants. As a result, nearly all the servants were sent home.

The same would have to apply to any "tag-alongs" who weren't carried on roll. Food, clothing and arms had to come from somewhere on a regular basis, or the child would have to do without.

If Austin really managed to tag along for any length of time, I would expect his name to appear somewhere in a wartime document. In this case, the letters should mention specifics about service -- camp experiences, marches, towns and places seen and comrades remembered. There may be some kind of document which lists him in the National Archives collection, "Unfiled Papers and Slips".