IF, repeat IF you are safe working around 240 VAC, you can easily check the ATS:
With generator off, shore power disconnected, remove the top/lid of the ATS:
There will be three sets of wires into it-- from generator, from shore power and to coach main AC breaker box. Each of these has 2 hots, 1 neutral and 1 ground.
Plug in/turn on shore power.
Check for power on both the IN from shore power side and on the OUT to breaker box side.
That will tell you exactly what you have/don't have.
NOTE: If, when you remove the lid you see burned wires, stop and determine if the wires can be cleaned and good connection made or if they have damaged the ATS itself.
If any of this is not fully understood and/or you are not safe working around 240 VAC, have someone qualified with this set up do the troubleshooting.