There is a relay that clicks when the ignition turns on and it is probably
completely unrelated to your problem.

I'm more of a believer in clean connections than in dielectric grease.
Either the grease is conductive or nonconductive. If it is conductive
it can result in shorts if it gets in the wrong place. If it is nonconductive,
why would you want it between parts that are supposed to be connected?
Yeah, you can put nonconductive grease over your battery terminals AFTER they
are connected and good and tight. But you don't want grease between the
post and the cable clamp.

Have you looked at the connection between the battery cable and the starter?
Also there is a ground strap between the engine and the frame that might
give you that symptom. And the far end of the battery ground cable.
Those are about all the places where ALL the electricity goes.
--
George Downs, Bartlesville, Oklahoma, Central US