You kind of mixed terms there in talking about the return campaign free trial. Those are two different (and mutually exclusive) programs.

If (as sounds more likely from your post) you're talking about the Callback Campaign, I don't see any reason why not.

If, however, you're talking about the actual Free Trial of the game, then no. From FINAL FANTASY® XIV - Free Trial Terms:
  • Free Trial Players cannot make in-game micro transactions during the Free Trial.
  • Free Trial account characters have their level capped at level thirty five (35).
Either of those rules would eliminate level jump potions, and the first one would also apply to story jump potions.