Aspect of the Beast says you can take it without meeting prerequisites if you have lycanthropy. The mooncursed barbarian's flavor text says you have a form of lycanthropy. Can they take the feat?
What about shifters prior to getting wild shape?
Not sure about the Mooncursed Barbarian.
A definite no for a Shifter of level 3- who does not yet have Wild Shape. There is nothing to suggest that they count as lycanthropes.
There's a precedant for non-lycanthropes counting as so with the skinwalkers, but I'm not sure if mooncursed counts.