Car acts dead after it gets hot...

So you are bypassing the relay by jumping between the two lugs.

Next time try to activate the relay by hooking up a remote start switch to (I don't have a wiring diagram handy) the yellow wire ( please someone double check me here) and positive battery

If it starts the relay is fine, issue is elsware, if it doesn't, ground the wire that would go to the NNS and try again.

If it starts, issue in NSS cir, if it doesn't points to relay.

That's how I would track down the root cause.