I'm not sure I'd agree Alice being able to use the cube to heal is entirely out of left field. We already knew from the fight with the snake guy that the cube can forcibly take in mana from multiple different sources and make them work together, presumably meaning it needs to convert that mana into natural un-assigned mana. Basically, it does the opposite of what happens when a living being takes in mana.
Presumably then Alice is going to pump her mana, along with Sophia's and Cain's, into the cube and then reverse the process again to transfer it from the cube into Eve. Not something we've explicitly been told it can do before to be sure, but it does seem like something that would be within it's abilities.