That's a weird one.
The first thing I'd do is to lift the tank and swap round the ignition and headlight relay as they're both the same. At the same time check that the terminals and sockets are clean.
Do you have any error codes logged? If "LIGHT" is shown, that indicates a problem with the relay.
Could there be a short in the wiring somewhere directing the headlight voltage to the high beam warning light? Out of curiosity, when the light isn't working, does it illuminate when you press the 'pass' button.
On the Evo, the light is programmed (presumably through the dash?) to extinguish 60 seconds after either killing the motor or turning the ignition on but not starting the engine. If your 1100 is the same, perhaps there's a problem with that aspect. Try disconnecting the multipin connector from the dash and cleaning the contacts.
Good luck!