Obviously the alarm module needs to somehow tie into the wiring going to the starter relay/solenoid. From your description the alarm module starter relay is tripping and that should send the power to the starter solenoid. Did your alarm come with a wiring diagram? Find the wire from the alarm that triggers the starter and trace it until you find the problem. You may even be able to run a new wire from the alarm module directly to the engine starter. This should be pretty easy to figure out if you have a diagram and can trace the wires. Once you find the correct starter wire you need to put a meter on it to verify power is leaving the alarm module, its possible the relay is tripping but not sending any power down the line.
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