A good way to lubricate these handles (not the lock keyway, though) is to open the boot lid, horizontal to the ground and, over a period of time, bathe the joint between the handle and the fixed part (with the two screws) with WD40 or similar penetrant. Note that this will NOT lubricate the handle mechanism, just provide an excellent path for the light oil (3-in-1 or similar) that WILL lubricate the handle. Bathe this handle joint over a period of time while the boot lid is parallel to the ground and hopefully sufficient lubricant will find its way through the joint to ease the handle movement.
See my separate post regarding WD40 and the dangers of treating it as a lubricant, which it is not (well, not very much, anyway!).