If the 3rd gear was jumping out of gear you should feel that clearly and the shiftstick would jank into a false neutral between 3rd and 2nd, so I assume that is not the case here. There is nothing different happening in the gearbox in 2nd compared to what is happening in 3rd, I mean the mechanic works is the same so there can't be something wrong with the gears or mechanics inside the gearbox if the 3rd is staying in gear. The 3rd is coupled via the slider gear dogs and splines, to the mainshaft that goes directly to the clutch, doesn't engage any intermediate gear as in 1st or 2nd. If the mainshaft would slip on the clutch hub because of a broken key, that would rapidly get worse. If there should be anything wrong with the chain or sprocket or splines that hold the sprockets on the shaft or hub, that should be felt in 2nd or 1st as well. So there must be something else wrong.
The only logic fault that I can think of is the clutch if there is no apparent difference at the start in 1st, 2nd or 3rd gear. Maybe you haven't charged the engine enough in 2nd to feel the clutch slipping, the strain usually is larger in 2nd, up a hill when the engine is revving on it's torque curve peak.
Do you use original plates, original style aftermarket plates or composite new material aftermarket plates? Do you use original or new springs, 16 or less? What kind of oil, brand, weight and what specification does the oil have?
On the rear stand when the engine is idling, engage the clutch with you hand. Can you feel a small slack at the pedal or lever just before the clutch is engaged? The free play should be small, about 1/2" at the tip of the clutch lever. Too big and it is a risk that the bearing ball ring in the clutch release bearing starts to rattle and can break apart with that. Too small and the bearing can gall and jam. Release bearing breakdown is very VERY dangerous.
When you progressively engage the clutch with your hand, do you feel any undulating or vibration that shouldn't be there? Is the engagement smooth? Is the de-engagement, the returning smooth? The worm needs to be packed with grease in many cases to work properly.
I have found that on my bike, the smoothest get-away position of the clutch lever is when at rest, the lever is keyed the farthest back that it can be put. That is because the slipping point, when the bike wants to depart, both the pedal and the lever at the same time is collaborating in the most favourable position for the touch and feel in the foot and I can portion the slip very precisely.
