You have a Leman rifle, a style sometimes called a Plains rifle. Leman continued to make muzzle-loading rifles for quite some time, so the 1867 could be an actual date, and those letters are cleanly engraved enough for them to have been executed by somebody with a bit of skill- not Frank James with a chisel. The top of the barrel, however, usually has the maker's name. An owner would be far more likely to have his name or initials on an inlay, very often a small oval set just behind the tang of the barrel, on the wrist. Leman was quite the manufacturer, and it's maybe a bit more likely that these are the initials of two store owners who ordered some rifles and retailed them, e.g. "Francis and Johnson, General Merchandise". It also occurs to me that two gunsmiths, "F&J" could have simply used a Leman lock to make this rifle. Here's an example of one like that. The F& J triggerguard is different style than the typical bow Leman used, and looks a bit like this Texas rifle's , no?

To link it to such a famous figure ( and considering that, as you say, James' mother notoriously passed off a lot of guns she said were Jesse's) , you'd need a better provenance than what you have right now. Unless you can actually come up with a paper trail, it's going to probably stay just another interesting story. However, Leman made a lot of rifles, and quite a few survived- late guns like this didn't get much use, as they were quickly obsolete. Though some were consumed in scrap metal drives in the two world wars, there are still a number of them around. You might start looking around for examples, because it's possible that "F&J" exists more than once. If so, you can decisively ditch the James story.