Gidday mate. sorry to hear of your predicament. If you can not find the lost fob, then you're stuck, unfortunately. I can't see any way round it if you don't have either of the 2 spare keys & no fob. You'll have to wait for the new one to arrive in 2 week. As for how the frequency of looking for the key, I did an experiment, I started my bike & removed the fob to inside the house & also turned it off. The bike kept running & no warning light came on at all, however, when I turned the bike off it knew there was no fob & stayed shut down. I looked online & AI came back with the below, so going by that, where your light came on or just prior, appears to be the likely place.
Good luck.