Your opponent can do what he wants (pro-Red his Avenger and destroy your Avacyn).
Both Avacyn's flip and Avacyn's damage are triggered abilities that go on the stack. As such, they can be responded to since both players get priority before they resolve. (See this questionthis question about how the stack works)
There is a small kink in this scenario: if your opponent is at 5 or less life, they also get pro-Red, so no damage is dealt to him/her, and Avenger can't tap to destroy Avacyn (yet).