I think a better way to salvage the heat is to pump that gray water through plastic tubing in the floor similar to a floor heating system. In my mind, such a system could never be used to heat a house, but would simply reduce the demands on the house's conventional heating system, thereby saving money for the homeowner.
New warm water could be added by a second higher pressure pump and a pressure relief valve somewhere on the tubing could dump exactly the same amount of cooler older water into the drain or perhaps a large storage tank for flushing toilets.
The problems, however would be in the practical considerations and the economics. Food particles and the cooking oil/water emulsion from the dish washer and soap scum from the shower/tub would cake up and clog the inside of the tubing. And, of course, buttons, lint, thread and the like from the clothes washer would cause no end of problems with the pump impeller section and cause blockages in the grey water tubing and the pressure relief valve to malfunction. Then, repairing those problems would eat away the accumulated savings.