Neutral light out and start button not working with everything else working strongly indicates a problem with the neutral switch and its wiring. Your test with the stand up and clutch lever in should have bypassed the neutral switch and allowed the starter circuit to work, but there could be additional problems in those switches. This should be easy to track down.
First do the easiest thing. Make sure the transmission is in neutral and check for the neutral light and for starting and leave it in neutral.
Next, pull the green neutral wire off the neutral switch just in front and below the front sprocket. With the transmision in neutral, check for ground on the switch plug where you pulled the green wire off.
Next, ground the green neutral switch wire and check for the neutral light and for starting.
If none of the above worked, we will test the small starter circuit relay. I have not had to mess with this relay on either of my gen2 KLRs, but I think I have seen it behind the dash. It is about as big as my thumb with a cylindrical end. If you have gone this far, the neutral light back there probably needs replacing anyway. You can get one at an auto parts store.
This relay should make a little click when you push the start switch with the key on. If not, pull the connector off the small starter circuit relay and test on the harness side of the connector for ground on the blue/red wire and then with the key on and the start button pushed check for 12 volts on both of the two yellow/red wires at the connector. If both these tests pass, the relay is bad. You can replace it or disassemble it and most likely fix it.
If none of this solves the problem, we can go farther.