Triton,
Anytime you turn on the S-7 Battle Short, you bypass most of the safety features on the gen set. So yes, you can force the K-1 output contactor to close. BUT, you also leave yourself in the position of destroying something.
I once went to a unit that could not get a gen set, (MEP-006) to put power out. Short Circuit idiot light came on. No power. They ASSUMED, this is the key word here, and will almost always get you into trouble, assumed that the short circuit relay was bad. They ordered one, but the ship to receive time was two weeks. That's too long, so they jumped the relay by flipping up S-7. Everyone was dismayed when the wire harness started to smoke. Didn't take long! So use the S-7 with caution. Its a wonderful troubleshooting tool, and in battle, I would certainly use it when push came to shove.
If you look at your wire schematic, you will find the S-1 circuit schedule. It will tell you which pins should be hot when the switch is in which position. And as you found out, the switch is in order. So now you need to start looking between L5 Fuel solenoid, and the S-1. Work from the L5 backwards. In the manuals section of SS, you will find a color coded schematic. Use it, if this kind of thing is new to you. We haven't any idea how much you know about electrical troubleshooting. Frankly, I have almost 40 years experience, and still use the color coded schematic. Its simply easier. This sounds like you have a stuck relay. I once saw a gen set that wouldn't work right, and when I pulled the relays out of the sockets, found one of the wire clips that hold in the relays, (something I feel is unnecessary) had become trapped between the relay base, and the socket. Shorted out to pins and that was the whole problem. Never know, so give it a look see. You can also change the position of the relays. see if you get a different fault. Then you can find the bad relay easy. The "relay" on the left side of the control panel, is not a relay. Its a switch. Different thing altogether. So dont mix it up!