Should I install vapor barrier in my basement bathroom?
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.