Are you saying you would wash your dishes using water from your fresh water holding tank rather than the hose connection? If so, how are you going to rinse the Calgon off the dishes? Calgon is sodium carbonate which can cause trouble swallowing and inflammation of the throat if ingested, or pain and reddening of the eyes.
You would have to be able to rinse the Calgon water of the dishes to avoid ingesting any of it.
Maybe look into getting an on the go portable water softener to attach to your trailer.
Chuck and Robyn - Camping since 1985
From tent camping to 2013 - Sunset Trail Reserve 32 Bunk House