Well, in theory, you should be able to.

HOWEVER, its not a terribly good idea...

To get the non-weathered version, you have to at least clear T7, and almost certainly not just once. Which means, at this point, you're doing Coil...and have a chance at your Allagan Ring (Casters and Melee get theirs T6 or 7, so definitely don't go buy two Sol rings as any of those 4 jobs) Now, the tank Ring drops T9, so if you are a tank this could be a viable way to go, since you're likely to not see your Ring for a long while, but the other roles...no, just wait for the Allagan.