Will a “whole house” thermostatic mixing valve installed at my water heater (hot water line) correct my pressure loss and ensuing hot and cold shower problem?
No, IT WILL NOT but it will allow tempered water to flow throughout the hot water piping.
Personally I would run a dedicated hot water line to any dishwasher Or washing machine THEN use a Holby Valve to temper the rest of the hot water.
Tempering valves work on temperature NOT Pressure
For pressure you would need either a pressure regulator Or a pressure balancing Valve
Or you can use a combination pressure temperature regulated valve
Or you can lower the temperature leaving the hot water heater and add a pressure regulator to the system.
Or you can fine tune the system by finding out why your getting a sudden frop in VOLUME that causes these temperature fluctuations