Some updates:
I ordered a Silicone Tube (8 mm ID, 12 OD) and cut 6 small pieces of 16 mm height.
I removed the plate, loosened the 6 screws and removed the heatbed. Then I placed the 6 silicone spacers around the original spacers.
I put the heatbed on top, tightened the screws again (but not completely), and then put the plate back on top.
Finally, I computed the heightmap 4-5 times, each time tightening or loosening 1-2 screws. The result is shown below. I went from a 0.8 mm min-max range to less 0.1 mm and the whole process took less than half an hour.
I now need to make sure that the leveling is robust and is not destroyed after a couple of prints.