I have an unused CH hat switch here. It is the thumb one from a CH Fighterstick which I removed immediately for another project. Assuming the post / shaft underneath the hat switches are identical (a reasonable assumption given the baskets seem the same) I measure the diameter of the shaft at the switch touching part to be 5.64 mm. This is an approximate measurement because I only have a Vernier Caliper and no micrometer.
I think that opening out the aperture at the top of the assembly will only be a temporary fix because the switches themselves will eventually bottom, thus this will limit the leaning travel of the shaft. If it is the case that these shafts wear, then that would explain the difficulties of using two switches simultaneously (for back left view, for example). This is supported by what DREDIOCK says.
I haven't had occasion to test this because my stick is still fairly new, but I was thinking that an alternate solution to trying to make the shaft wider would be to glue something thin to the surface of the switches, the little grey cylindrical part that sticks out. As long as there is a little backlash (slack) still on the switches when the hat is in its neutral position it should be alright.
You could measure the shaft and compare it to my measurement and compensate any reduction by a similar sized spacer on the four switches. It would be a lot simpler than widening the shaft since the switches have flat tops.
Also, you can rotate the actual hat and therefore shaft in 90 degree increments once you have removed the screw from the bottom. Perhaps turning it 180 degrees might be a prudent measure, assuming you check behind your aircraft the most.
Just a thought anyway, worth a try if the thing if f.u.b.a.r. already...