Crave on any character seems to work in reverse with Goddess Asset healing.
Whether it's Asset's Fury or Power healing ability, if she uses it, any ally currently with the Crave buff gets HP drained not added.
It took me a while to figure out why her fury was killing allies. But after watching closely I seem to be able to confirm that Crave causes Asset's healing to work in reverse, removing HP rather than adding it. In fact, she's even killed herself with her fury due to this.
Note that other characters can heal with the Crave buff present and things work as expected. it's ONLY Asset's healing that seems to be broken with the Crave buff.