OK, I have a photo of my '06 12X fuse/relay layout with the start relay sockets marked. Please forgive any 'stating the obvious' remarks - just trying to cover as many bases as possible. :)
When the starter button is pushed +12volts is provided to the start relay under the seat. In the picture below (taken from the LH side of the bike) I have removed the start relay and identified which sockets connect the relay winding to power and ground.
[Depending on how thick your multimeter probes are you may have to use something like a resistor or capacitor lead, small diameter wire or similar to get contact to these sockets.]
With the ignition and kill switch ON you should read +12volts (battery voltage) on the 'POWER' socket when the start button is pushed. If not, then the wiring to/from the start button on the RH handlebar needs to be checked.
With the ignition OFF you can measure the resistance from the 'GROUND' socket to the battery ground and depending on the type of meter you have you might or might not get a resistance reading because the path to ground is via that diode that is visible on the far left of the picture. If you pull the clutch in you should see the resistance drop to something like 0.5 ohms as the clutch interlock switch provides a ground.
If you are getting this low reading with the clutch pulled in, the one test I would do is to try starting the engine with the clutch pulled in (even if you are in neutral) to eliminate any issues with the interlock diode / neutral switch (obviously with the relay refitted). In your post you mention checking the clutch switch but I'm not sure that you have tried starting with the clutch actually pulled in?
I had intermittent starting issues on my '06 ages ago and it turned out to be the neutral switch being resistive and not allowing the start relay to hold in once the starter engaged. You issue sounds a little different in that it doesn't seem to be attempting to activate the Start relay.
Anyway, this should get you started - the fact that you can get the starter to engage by bypassing the start relay contacts does narrow this down quite a bit.