Took the plunge with drew e link and went through the password process to get some of the info.
so the wiring diagram you need for spec k is 611-1267
By the looks of it fuses are not there for 12v side.
the breaker on the gen is only for 120v as u probably already know that.
on the diagram it looks like J1-8 on the pinout is bat+ J1-6 is bat-
J1-16 is starter wire that goes to solenoid that allows starting to happen.
looking at the diagram you need 12v on the large lug on solenoid and you also need 12v+ on the small pin on the solenoid while push the start button. this energizes the coil and ignition
With a test light lead connect bat+ and check ground on outside of frame lug. I would also check if possible J1-6 with a pin or paperclip to verify results that you have good ground light should come on if it doesnt BAD ground.
next would be move test light to bat- and check solenoid which i think you did. i would also check the signal wire or J1-16 which is small pin on solenoid WHILE push the start button if light comes on bad solenoid if nothing problably bad board as long as grounds check ok.
Then I would call the board now there also could be pin fitment issues as well as corrosion so i would take a careful look at the J connector on the board and the pinout to rule this out pin fitment can be as simple as unplug replug or spray contact cleaner if you see any green crusties. as your board is 327-1413 and is approaching 500$
your solenoid is 307-2586 theoretically you could add a small 10amp fuse to a jump wire and jump from bat positive to the small wire on the solenoid to verify your starter turns this might even be the quicker way to verify board however pin fit and corrosion can still be the wild card.