Well, you're not going to be happy if they order you a shift shaft because that is NOT your problem, if your description of how it shifts, is accurate. If you can go from 4th gear to 6th gear and back to 4th without a problem, your shift shaft is doing its job. It does not know what gear you are in, only that it has to turn a drum, and if it is able to change ANY gears up or down, it will change every gear up or down, unless there is a problem with the drum itself. If the guys at the shop don't realize this and are ordering you a shift shaft, they need to take it apart first and see what the real issue is. And if they're the ones telling you all the horseshit about how a shift shaft gets bent, or forks get spread from pushing on the shifter too hard, (they can't) etc , you need to take that bike somewhere else. There are pins on the end of your shift drum that the shift shaft needs to "hook" on and turn to up shift or down shift. Maybe the pins holder on the end of your shift drum has come off and the pin the fork needs to hook to shift you into third has fallen out, or if it is part of the drum, broken off , but it is not the shaft itself. There is a spring that could be at fault, but, the way you describe the problem, it would only return in one direction, so you could upshift, or, downshift, but not both. I can't believe this shop is special ordering you a part without diagnosing the problem first. At any rate, I would do what EVERYONE here has suggested, regardless of how it "feels" and grease the pivot. If that still does not work, I would take the bike to a dealership that knows what the hell they're doing.
This edit is a couple of hours after I posted the above. I know it's not smart to post something with such certainty, less the one in a million chance that something I have NEVER heard of or SEEN in my life has happened with that shift shaft, LOL.
My bigger concern is that shop, orders you that shaft, charges you for it or warranties it, finds out it is not the problem, holds your bike longer while they figure out what the real problem is and order more parts if needed, and then TELLS you it was the shaft AND other parts, just to save face.