I have had the exact problem with a 2006. The transmission simply would not shift.
Doc,
Please look at the following rationale and tell me if I'm full of fertilizer or not.
1. When this happens I don't feel anything, I mean nothing.
2. In theory, you don't have to use the clutch to shift up or down.
3. If you pull up and press down again, let the clutch out and then back in, sometimes it works.
4. If you shift one at a time, "most" of the time it works.
IMO, this smells like the shifter mechanism. I haven't looked at the FJR mechanism but most of them are some kind of ratchet affair with notches or something to allow some type of indexing. I don't think anyone since the old X6 Hustler has tried a circular shift patter (going from 6 to 1 got exciting). It just feels like the lever isn't moving the ratcheting mechanism like when a socket wrench just sits between clockwise and counter-clockwise.
Does that make any sense???