That is clearly a No. 3, First Model, aka, First Model American. The use of a nickel for a front sight replacement was (and is) common. I have done it, though I flatter myself that I did a better job.
The serial number probably was 1911; those guns only went to about 8000 and S&W always stamped the serial into the right stock. That was not to keep collectors happy, it was because they polished the grip and stocks down together and wanted them to get back together after final finishing/plating was done.
However, the original serial was apparently removed at some time; it should be on the bottom of the grip, upside down in Picture #5. I think I see some signs of it, but can't be sure.
I don't see the U.S. marking, but I might have missed it. But the Army guns were in the 125-2199 serial range, so that one is right in that respect. Further, it is one of only 200 of the 1000 gun order to be nickel plated. It would have had OWA (O.W. Ainsworth) on the left grip. "A" and "P" markings are also seen, apparently government inspector marks. I don't see a mention of "G" or "O" marks, but that means little.
So the gun would be pretty uncommon and valuable even without the Frank James association. I doubt the "stolen from Springfield Armory" story. Springfield Armory is in Massachusetts, and AFAIK, the James boys never got that far east. Of course they might have received stolen property but it seems more likely that the S&W's were sold off by the Army and could have been obtained by about anyone. (No gun laws or background checks, remember?)
IMHO, a Frank James association would be hard to prove or even assert seriously based on nothing more than a family story and a couple of scratched-on letters that might or might not be the letters "J" and "F" in the wrong order.
Jim