The Second Solenoid is probably the FIRST solenoid, your House Battery Disconnect. It's got to be making up to "some degree" because most of your RV's 12-volt-DC stuff seems to be working. It could be damaged to where it won't pass enough current for the Genny, or (Again!) some Connection. Does it look sorta like This?
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You can safely jump across the two LARGE terminals to eliminate the Disconnect as your problem.
Disconnects are Latching Relays, different than your Starter Relay. Latching relays stay "in state" once their control signal goes off. Your Start Relay has to have its control coil powered up, to continue cranking.
If you determine that you need a Disconnect Relay, know now that there are at least two different designs. Depends on the type of Control Switch. So, don't jump into an order. We can help figure out which type you need, but it's less than straightforward.