I just thought of something else

Henry said that the leader is a great man, but you should not cross him.

Who did Ana Lucia cross? Christian Shepard. (Well, she's crossed a lot of people, but...)

Was Michael's mission specifically to kill Ana? (In addition to bringing the
Losties to the Others?)

Was is given to him by grand poohbah Shepherd himself?
___________

I don't know if it would be an effective mission to get Henry out, because
Henry probably figures that he's dead if he goes back to the Others, so I'm
leaning more toward his mission being to bring the Losties to the Others in
whatever way possible.

Obviously the Others know about Ana, because she killed Goodwin (and Henry
knew she did), so if his mission was "bring us most of the survivors of
Flight 815 and we will give you Walt back" (which seems stupid to me anyway,
because even in a best case scenario, Michael and Walt would be ostracized
by both groups), part of the mission could have been "if you get a chance,
kill somebody, here's an approved list of victims," and Ana would have very
easily made that list.