Looking back, a couple of players' trajectories might be relevant... Kyle Connor and Jack Roslovic.
D+2 seasons in the AHL:
Roslovic 0.74
Connor 0.85
Lambert 0.86
D+3 seasons:
Roslovic 31 games Jets (14 pts), 32 games Moose (35 points)
Connor 76 games Jets (57 points), 4 games Moose (5 points)
Lambert TBD
I'd assess Lambert to be ahead of Roslovic and behind Connor in terms of development and production, at this point. Notably, their D+3 seasons (when they both got a considerable run at the NHL level) were in the Jets' big contender season, so they weren't exactly in a "youth" movement.
So, I wouldn't be all that surprised if Lambert is given a good run with the Jets next season if he shows well in training camp. He might be up and down a bit, but it will depend on trades. One thing I would like the Jets to do is to push Appleton down to the 4th line and try another skilled RW with Lowry and Nino. That line could offer more offense if they added skill to RW the way Jarvis does with Staal on the Canes.