“This past weekend I was looking to rig up a filter for the latest round of the Samsung washing machine battle.”
You may need to put the phosphates back into the detergent. The modern machines use less water, colder water, and detergent that is just dyes and perfume. If you do try TSP or other phosphate, you may have to use hotter water, and a lot of phosphate detergent. The synthetics accumulate body oils, then the oils go rancid and hard, meaning smelly and hard to remove.
Getting badly rancid oil soiled laundry clean may take a trip to the dry cleaner. Then start using TSP with your detergent.
Government ruins everything.
We live in an area where water is comparatively cheap. So instead of harsh chemicals like phosphate (which harms the water table and local fisheries), we pre-soak each wash load for about 40 minutes, adding powdered Oxy bleach to the white load, or borax or a capful of Lysol or Simple Green to cut grease if needed, and completely drain off the soak water before refilling and running a wash cycle. Works great.