In　this　paper,　we　study　static　bistability　and　mechanical　cooling　of　a　dissipative　optomechanical　cavity　filled　with　a　Kerr　medium.　The　system　exhibits　optical　bistability　for　a　wide　input-power　range　with　the　power　threshold　being　greatly　reduced,　in　contrast　to　the　case　of　purely　dissipative　coupling.　At　the　bistable　regime,　the　membrane　can　be　effectively　cooled　down　to　a　few　millikelvin　from　the　room　temperature　under　the　unresolved　sideband　condition,　where　the　effective　mechanical　temperature　is　a　nonmonotonic　function　of　intracavity　intensity　and　reaches　its　minimum　near　the　turning　point　of　the　upper　stable　branch.　When　the　system　is　in　the　cryogenics　environment,　the　effective　mechanical　temperature　at　the　bistable　regime　shows　a　similar　feature　as　in　the　room　temperature　case,　but　the　optimal　cooling　appears　at　the　monostable　regime　and　approaches　the　mechanical　ground　state.　Our　results　are　of　interest　for　further　understanding　bistable　optomechanical　systems,　which　have　many　applications　in　nonclassical　state　preparations　and　quantum　information　processing.　©　2023　American　Physical　Society.