MacManWork, it seems that by bypassing the solenoid and having it crank, you are pretty much troubleshooting the problem to the solenoid. Although the solenoid could act similarly if it is not getting enough current to activate properly. Starting your rig first as prior post suggested should provide the proper current. The activation (small wire) terminals on the solenoid could also be not making good contact. I would try cleaning those as well. I don't think any of that is easy to get to in that genset, however.
If you have the genset switch set to "Run" and you can crank it by bypassing the solenoid as you did, it should start and stay running.