fuel cut relay

Where did you tap the ignition wire?

A relay is simply a resistive coil.  If you put either 12vdc or ground on one side of the coil (pin 85 or 86) and leave the other side of the coil floating (not connected) you will read the exact same signal on both coil wires (the signal will bleed through the coil as long as the circuit isn't completed).  So, to test your (-) when armed wire you would need to pull the relay out of the socket to remove the coil.

I can't see why the car would start fine with the key but not with the remote start.  Unless you tapped the ignition out of the remote start system.  Even then, it should still work unless your ground when out has been damaged (output a - at all times).

I don't think moving the wire will help.

When you try remote staring the car see if you have 12vdc from pin 85 to pin 86 of the fuel bypass.  If you do, then you will need to cut one side of the relay (say the orange side) whenever you remote start.  You can do this by connecting the ignition wire to the relay and the (-) status out (should be small blue) to the other side of the second coil.  Then, wire the orange wire going to the coil of the first relay to pin 30 and install a jumper wire from pin 87A to the coil of the first relay where the orange wire used to go.

Does it sound like the car is starving for fuel?  Try bypassing this cutoff and see if that allows you to start the car remotely.  It doesn't sound like you are confident that the fuel cut relay is what is causing your problems.  The first thing you need to do is identify the problem, then you can work on fixing it.

So, to identify the problem, eliminate the relay (wire pin 30 and 87A together to bypass it).