Pretty much the problem with the faucet and hot and cold water mixing is the water heater usually has a check valve on the outlet and a minimum supply of hot water.
As long as the shower is flowing both hot and cold water are flowing. When using the shower head shut off the cold water continues to trickle and the check valve on the WH seats since the flow thru the WH is now restricted. When the flow starts the hose is full of cold water and the hot water flow has to catch up.
The other problem is that as you use hot water from the WH the water in the WH becomes colder and colder. So the only way to fix this would be to have some sort of temperature regulated mixing valve set up that can adjust the cold water flow to the shower head, but you still have that limited supply of hot water that ain't gonna last.
I myself just keep on reaching down and cutting back the cold water valve as needed and take navy showers.
I haven't seen any all in one faucet/temperature mixing with shower hookup with 4 inch centers. Sink faucets only, yes. You might look at adding a separate through the wall shower hose/line connection and adding
separate mixing vale in the plumbing to supply the separate shower line.