Rob,
The contactor that you pushed in has a coil that is energized to pull the contactor in. It could be as simple as the coil is bad on the contactor, or the transfer board is bad, or a bad connection.
With a multimeter check to see if voltage is present on the contactor coil wires. There should be 2 wires on the contactor that will be energized if the transfer circuit board is working when the generator is proven running.
This link below may be for your system, it may do a better job describing what I attempted above.
http://www.progressivedyn.com/pdfs/110537%20English.pdf
No power to loads when generator is running:
1. Disconnect Shore Power.
2. Start the generator, measure AC Volts between HOT1 and NEU on the GEN connections, there needs to be voltage
between 105 and 130VAC to energize the contactor. NOTE: there is a 20-45 second delay before the generator transfer
contacts engage. When voltage is present between HOT1 and NEU the green LED on the control board will be
illuminated. If the proper voltage is not present then there is a wiring error between the generator and the transfer
switch or a generator problem.
3. If proper voltage is present at HOT1 and NEU on the GEN side wait 20-45 seconds then measure AC Volts on the
load side between HOT1 and NEU, it should read between 105 and 130VAC. If the voltage is not present and there is
proper voltage on the GEN side the transfer switch is defective.