There appears to be an enmity component to FATEs. You have to produce enough enmity to be noticed by a mob. As a healer, this means overhealing. Not just topping off, but overcuring. You need to be able to produce as much enmity as a DD in a fight, or more. Raising also seems to produce enmity. My sister does this on Odin/Behemoth, and it at least seems to help her get silver. Overheal people who have enmity from the big guys, and raise people when you can and heal 'em up.

It is horrible that this is this way. Supposedly for 2.1, SE will be adjusting things for Odin/Behemoth to make them work better.