Tom,
before everyone bombards you with thigns to check, what have you checked?
starting problems usually start ( or dont start as the case may be) with the battery, and its terminals.
then if the battery terminals are tight and the battery is in good condition making full power there are other areas to check.
on the Rocket, the clutch must be pulled in so that sensor can go bad or get corroded, and the same for the sidestand sensor.
also grounding issues, as well as starter solenoid issues. ( corrosion on the contacts for the starter etc.
Id check all of those things before even attempting to diagnose any other areas.
the actuall starter button itself or the relay could also be bad or corroded.
(easy to check if the other areas pass inspection)
I know your UK weather is tough on bikes.
the noise you hear under the bear claw, could be the electrically controlled secondaries being activated. the fuel pump is in the tank and the Fuel pressure regulator is along the right side if memory serves me.
( I get confuesed with the sprint and tiger I also have worked on sometimes)
there is also a fuse under the seat, but if the fuse were blown it would not start at all. and if it was not groundign properly you'd have more issues startign when hot, not cold.
have you downloaded the service manual? its a big help and available on this site if you look under any post by Bishop ( Marcus)
good luck!!