You can redo the MSQ on the same character via New Game+, but you won't gain exp. So if you want to level Reaper while doing the MSQ, then you'll have to do it on a new character.

And to unlock a Reaper, you'll have to have a job at level 70+ and be able to access the starting quest location for Reaper. You can use boost to level 80 or level another job manually to level 70.

There is currently no other alternative.