You may not need to actually move the vent. In the picture the greenboard is against the studding and the back of the shower pan was installed against the green board, therefore the vent should be clear of the actual shower pan. You then have a second layer of wallboard on top of the greenboard which extends over the lip of the shower pan about 1/2".
Consider installing a cripple stud on each side of the opening leaving the face edge of the cripple 1/2" proud (further out) of the existing stud to compensate for the greenboard layer. That will give you another 1/2" front to back clearance and you should then be able to bridge the new wallboard across the opening and clear of the vent.
It is difficult to tell what they used on that base, but given that you say it crumbled i would suspect the tile installer used grout to form a bullnose. The problem with that is that grout is not waterproof so if it is not then properly sealed, in time moisture will be absorbed into it and cause it to crumble. Once the bullnose crumbles moisture will then be absorbed into the base of the wallboard and cause additional problems.
When you install the new piece of wallboard make sure the bottom edge of the wallboard is about 1/2" above the shower pan and allow the bottom row of tile to extend over the edge of the wallboard. That minor elevation of the wallboard will prevent the wallboard from leaching up any moisture that may get into that cavity.
Whatever that material is, the fact that it is crumbling indicates it is not holding up well in this application. I would consider removing it all the way around the enclosure, then perhaps install a strip of ceramic bullnose and caulk it all with a good TUB/TILE silicone caulking.
