I don't see that dog clutch being the source of noise.
That dog clutch is rear output in high range. If there was a funny wear that would cause a jump out of gear it would be in 2WD and not 4WD. That is because the interlock pills will lock the rear output in place while the front is engaged (you can't nave the rear in neutral with the front in gear, in stock trim).
Little reminder since it shows up a time or two a year. Those teeth go into the transfer case when you put it together. The gear will go onto the shaft either way. Put it so the teeth face forward (toward transmission) and everything will fit fine. But you will have no rear output in high range (2WD or 4WD), and front output and all of low range will work perfect.
I would still be inclined to reuse the gear.
When you are done, there will be noise in low range. That is a given. Those gears are straight cut. Straight cut gears are noisy, no way around it. But that is only low range and it isn't real bad.