If you wash it with the bottle detached water will get inside bladder. That will be next to impossible to dry out and will probably grow stuff over time if it doesn't it certainly won't do the firing mech any good when you replace it...
I've used vanish oxyaction with good effect on mildew on buoyancy aids. Make a paste. Work in to fabric with dish brush. Leave for an hour rinse off in bucket or with hose.