My proposed solution is simple, re-tune the undertuned exp sources:
+ 100% more exp from One-Time Quests.
+ 150% more exp from Battlecraft Levequests.
+ 50% more exp from Kills.
+ 200% more exp for Dungeon completion, encourage people to queue for in-progress and to stay.
+ 33% of the first-time bonus exp for Guildhest completion, for repeat visits.
+ Add Guildhest exp and gil for Trials, to encourage players to do them more than exactly 1 time.
+ Some large amount of Gil for Castrum Meridianum and Praetorium, to encourage players to do them more than exactly 1 time.
- 20% less exp from FATEs, they'll still be fast and effective, but you won't feel obligated to drop everything to chase every FATE that appears or grind only on them.

Make the investment of time in Quests, Levequests, Dungeons, etc competitive with FATEs, while also reigning in FATEs a little. It's MUCH healthier to have a variety of similarly efficient exp methods.