Washer water goes in to black tank.

as others have said you should not have to open the drain to use the washer no no no . these tanks are not sealed ! they have a plastic pipe that runs from the top of the tank to the roof, the Vent. Something is wrong with that piping! If its blocked it will also will cause odors to fill your coach. Fix the piping.
As far as plumbing washer to black tank .. My black tank takes forever to fill compared to the grey .. I would welcome the washer going into that tank for reasons already stated.