We study a war-of-attrition bargaining over a pie with heterogeneous parts, where players have incomplete information over their opponent preferences as well as their behavioral types. Each player demands that the opponent chooses from a menu of offers. The menu may consist of a single offer. If the bargaining position is exogenously fixed, we show that the equilibrium behavior can be simply characterized by comparing appropriately defined strengths of the two players. The equilibrium is unique with one-sided uncertainty, but not necessarily with two-sided uncertainty about preferences equilibria. We also consider the menu choice game prior to the bargaining. Being able to commit to a menu instead of a single-offer removes a certain first-mover disadvantage. When the preferences of one player are known, in equilibrium, this player proposes a menu of all allocations that give him a half of the pie; the opponent chooses optimally from such a menu.