[Second Series]

I am not an automotive electronics specialist, I am an electronics technician interested in automotive electronics.
I have read that a remote relay can be used to help fix the hot start problem where the starter is sluggish, or won’t turn when hot.
The typical starter has a terminal for the battery, and a terminal for the switch. When the switch terminal is energized it pulls the plunger that does two things. The plunger connects the battery to the starter motor, and engages the bendix. When the key is turned it energizes a coil, a large current is required for an instant to get the plunger moving, once the battery and starter motor connect, a second coil holds the plunger. The current in the first coil decreases for the duration of the cranking.
There is no way to not use the first coil, since it is required to get the plunger moving to connect the battery and starter motor. This leads me to believe that the hot start problem is caused by two things. A hot starter solenoid, and reduced voltage at the switch terminal. The reduced voltage at the switch terminal is the constant, it is only a problem when the solenoid gets hot. Since the solenoid can not be relocated, the wiring can be modified to eliminate the voltage drop.
This is where the remote starter relay can be used. Assuming the voltage drop is caused by excessive length of wire, from the battery through the firewall to the ignition switch, back through the firewall to the starter solenoid switch terminal. Instead, wire from the battery to the relay to the starter solenoid switch terminal. Control the remote starter relay with the wire that was connected to the starter solenoid switch terminal. Leave the battery cable from the battery to the battery terminal on the starter solenoid. This method would require a remote relay capable of say 40 Amps like the cube style.
Another way would be to use the remote relay on the battery cable, have one cable going to the battery terminal on the starter solenoid, and use a jumper between the battery terminal and the start terminal on the solenoid. This method requires the hold current and the starter motor current to go through the remote relay contacts, so the 200 Amp can style.
I modified your drawing on that part of the circuit to show two ways you could wire it.