It is fairly common that starting problems on the FJR first show up when the engine is hot. There are several theories as to why that happens, but no definitive ones. Could be the engine compression increases when warm, or the starter motor magnets becomes weaker when jheat soaked, or any of a number of other hair brained ideas.. Suffice to say that it is harder on the starting system when the engine is fully warmed up up than when it is cold.
That said, the first thing to rule out is the battery capacity. The 14V you measure at the battery with the bikle running is the charging voltage from the alternator, not the battery internal voltage. But 14V is a good charging voltage so at least we know that (with a good battery) it should maintain a full charge as you run the engine.
Measure the battery voltage before you start the bike. It should be 12.6 to 12.8V DC if the battery is taking a full charge.
Attempting to measure the voltage at the battery while actually starting is fairly pointless, since we already know that the voltage is dipping too far. The dipping voltage can be caused either by a bad battery (low capacity) or by a bad starter motor (pulls too much current), so measuring that voltage doesn't tell you much.
Instead,.you can get the battery load tested independently at a shop. The load tester will put a known load on the battery and pull a specific amount of current (amps) and they will measure the voltage at that load to see if the capacity is up to spec.
But given the consumable nature of batteries, if your battery is more than about 4 years old, you might want o just spring for a new battery to see if that resolves the problem. The next most likely problem is the starter motor is at fault, and that is a far more involved process to access and repair.