Water – Renovating bathroom, do I need to add a moisture barrier over insulation

Any high humidity area like a bathroom should have a good vapor barrier under the sheetrock. This barrier should be taped at the seams and ceiling forming sort of an air tight cell. Usually a kraft backed insulation is not a very good vapor barrier and has lots of gaps. To be on the safe side and avoid any condensation between the 6 mil and kraft, you could simply remove the kraft backing or cut several slices in it top to bottom. Usually it will not cause a problem unless you live in a very wet humid climate or the plastic is not sealed properly. The 6 mil is a much better material for your purpose than kraft backed.

No, there's no need to install a vapor barrier between two conditioned spaces. The purpose of a vapor barrier is to prevent warm moist air from traveling through the insulation and condensing on the cooler side, which can easily happen in cold climates. With a conditioned space on both sides, you can allow any moisture in the air to pass through.

There's also no need for insulation except for sound proofing (fires can pass through doors, as can any heat). For sound, make sure your insulation is designed for that, and also consider a resilient channel or similar product that will keep the drywall physically separate from the studs, which is how many sounds travel through walls.