Welcome to the world of Tigers. Both your theories are plausible. The heat
shield is obviously easier. Perhaps you could measure the travel of the
clutch fork when cold and then hot to see if there is an obvious difference.
If there is no difference then it seems the problem is internal. As you
stated a clutch that engages very near the end of it's travel likely has a
short life left. Ironically that aspect would seem to decrease the
likelihood of the problem you are having. You might also want to check the
attachment at the pedals and the adjuster at the slave. I have known
situations (though not on a Tiger) where things have been loose and the rod
can drop and wedge not centered in the fork cup creating a condition where
the clutch didn't fully engage. Later when the rod did work it's way into
the cup there was too much slop and it acted to not fully disengage. These
were very work parts (adjuster rod/clutch fork) that intermittently would
wedge and un-wedge and drove the party nuts.