Ah, the problem that always plagues most MMOs with Dungeons. Too few tanks and healers, too many DPS.
As someone who focuses on tanking and healing in FFXIV, my suggestion for the problem is this. If you run a roulette as tank or healer at max level, you'll get an XP-pill as a reward at the end, containing the amount of XP you would have gotten if you did run the dungeon below max level. After the dungeon is done, you then swap to a DPS job, pop the pill, and all the XP would go to that job instead of your tank or healer.
Of course, there would have to be some limitations to this. Like that you can't use an XP-pill on a job that wasn't high enough to begin with to do that dungeon.
For example: An XP-pill created in "Dusk Vigil" (lvl 51) can't be used on a Bard lvl 32 because that job wouldn't have been able to do that dungeon. But if you have one pill from "Brayflox's Longstop" (lvl 32) or "Halatali" (lvl 20), they would work on the Bard since these dungeons are within its level range.
These XP-pills should also only be available from Roulettes. Otherwise, you would have people spamming specific dungeons only.
Or perhaps this is a silly idea?