I have changed our transfer switch since we have owned our unit, so I have a good idea of how they are arranged and wiried on our unit and most of them work the same.
When you pull the cover off the transfer switch it will show 3 basic feeds
The incoming shore power
The output to the house panel
When you are plugged into house power you will see power on the shore power input terminals and power on the house panel output terminals. Depending on where and how you measure it will be 110/220. Nothing on the gen power. Usually the shore power feed you can follow from the cord. The Generator will normally come in from the other side. The house big cable will be the one that is common to both input relays on the output side.
When you unplug the shore power and start up the generator you will see power on the generator input terminals and again on the house power terminals provided the relays are working properly.
Now when I had my problem I couldn't get any generator power inside with a volt meter plugged into an outlet. With the gen set running I had power 110/220 coming out of the generator and verified this by reading on the generator input terminals of the transfer relay. When I looked for the transfer out of the transfer relay on the generator side I did not have it. Ended up being the transfer relay on the generator side. Changed the transfer switch and haven't had any more problems.
That being said, you first step is to verify you have the proper power in and out of the gen Set and the shore power and both relays transfer power to the house panel in their own operation.
Now if all the power is good at all locations, your job will be to idenify the small wires also attached to your transfer switch which feed the inverter for normal charging. If you find the one for the shore power which you say is working then you should be able to find the same location on the generator transfer relay that does the same function. If the power is there and leaving from the Gen set side you will have to trace the wire to find out why the power is not getting to the inverter input from the Gen set side.
Remember the input wires will be the big # 6 wires and the control wires for your inverted will be some what smaller.
I hope you can find it. It's a great feeling when you do.