this neutral light is a big thing with gsxrs from the srad up. both srad motors i had would do it now and then. my 08 1000 does it now and then. ive even seen brand new ones do it. that light has nothing to do with the fuel pump though. no matter what gear its in. kickstand up or down. clutch lever in or out. when you turn the key and flip the kill switch the pump will prime. the only things that will cause your pump not to prime are the tip over sensor, not having the pump plugged in (yes ive done that) faulty kill switch, blown ignition fuse or main fuse, faulty wiring in that circuit or the fuel pump wiring, or a dead fuel pump. of those the only things that would cause the starter not to spin are the blown ignition or main fuse, faulty switch, or faulty wiring. i believe the starter should still spin with the TOS stuck
to bypass the tip over sensor take it apart-inside theres a metal ring that slides over a metal rod. put a zip tie on each side of the metal ring so that it cant slide or contact the metal rod it slides over-take it apart and youll see what im talking about. You can also fill it with jbweld or silicone.
i actually had this very thing happen to me the other day on a ride. stopped at a natural spring outlet for a drink. came back out, key on. kill switch on and no pump prime or starter spin. it was my ignition fuse