You said the transfer switch clicks when you start the generator.
Does it click when you hook up to shore power? If it does not
you may have an energy management system, like I have in my coach, that will not allow shore power to the coach IF any of the following conditions exist:
NO GROUND
LOW VOLTAGE
HIGH VOLTAGE
NO NEUTRAL
REVERSED POLARITY
Just because you can read voltage at the power pedestal, that does not mean that your EMS is seeing everything as being correct. It will not allow it to switch to protect your coach from something that is wrong. Remember, this is what your coach is seeing at the end of the power cord, which may mean you have a defective cord and NOT a problem with the shore power outlet.