[franken]

Define "sometimes it won't turn on the starter when it's hot." Cranks slowly or no crank? Those are two different issues. If no crank does the solenoid click?
Ground the battery to the engine block via the big negative cable, not to the frame, head, bracket, etc.
Measure the battery voltage to the nearest tenth of a volt when cranking and with the key off.
Measure to ground at the starter on the big cable while cranking. Make the same measurement at the lug from the solenoid to the starter.
Scrap the remote solenoid. It's a waste and only makes for more parts to troubleshoot.

The starter runs on the high current side, meaning the two big battery cables.
The solenoid switches the starter on and engages the Bendix. It runs on the low current side.