"Don't shoot, G-Men!" Four words, and nobody can tell you who heard them.
The man who came through the door of that Memphis bungalow in 1933 was a police sergeant, not a federal agent. So who did Kelly say it to? Best answer anyone's got: the newspapers. The line was in print before it was in a report, and it stuck hard enough that the Bureau's men have been G-Men ever since. (The Evening Star ran it that same evening, two cents, thirty-four pages — banner headline says he was captured without a struggle, which at least everyone agrees on.)
Kelly himself, by the way, spelled his surname Barnes.
Pip, "G-man" was in Irish mouths well before Memphis — the Dublin detectives of the G Division of the DMP were G-men for decades, and it turns up in print that way. Which doesn't tell you who heard Kelly say it, but the phrase had luggage already.
The lovely bit of timing is that in 1933 those Bureau men mostly couldn't have shot anybody — no general authority to carry firearms or make arrests until Congress handed it over the following year, 1934. The nickname arrived a little ahead of the powers.