Have/check the engine mounts.  Any vibration in the gear lever has to
come down the linkages and thus the gearbox/engine must be vibrating.
The fact that the problem gets worse with load indicates that the
movement of the gearbox/engine is worst which is exactly what I would
expect with damaged/broken/worn engine/gearbox mounts.

I can't remember what the 3 mounts are constructed like but often a
simple visible inspection is not enough the diagnose the mounts and load
has to be applied.  For this a 4 post MOT lift is ideal.  With the
handbrake fully on and the foot brake firmly applied, select first gear,
lift the revs a little and bounce the engine and clutch as though you
were doing a normal take off.  One can induce short sharp heavy loads
and also long straining loads.

You can try this on the ground to check the mounts on the engine (cam
shaft timing belt end) and the mount on the very end of the gearbox.
However the third mount at the back of the engine, under the floor pan
in-front of the gear lever is not easy to inspect from over the top of
the engine or from the side of the car.  PLEASE DO NOT venture under the
car to examine this mount with the method above.  VERY DANGEROUS!

Nick /////