I just used a piece of 3mm Perspex I had lying around cut down to a small square to fit the plastic body of the joystick.
The hole I drilled was with a 19mm hole cutter. I didn't alter any of my mappings, they still work fine. However I just play the classics mainly and don't bother with analogue games. If you make a restrictor like this you WILL affect analogue games because the Ultimarc software compensates the maps for their original restrictors only.
I reckon you could go as small as 17mm hole, though I haven't tried this yet. At 17mm you probably might need to start editing the maps, and centering could be a problem. The plastic part on the end of the stick is about 13mm in diameter so a 17mm hole will only give you 2mm throw at the base. It might work, it might not. I'm happy with the 19mm hole I made.
Ahhh. Mine's about 17. It was done with a laser machine, which cuts in 'sections', so it's not perfectly round. It is about 17mm, so yeah, there's been some issue.
Ultimately that small means you can't use any of the center large box. Because of the inherent play in the shaft/base interface it can fluctuate as to which set of directions it'll favor - ie: currently it's at up-right, but before that was down-right, and it just changes whenever it feels like it. (I've documented this.)
So, you can, but you have to alter your maps every time it decides to change. I don't have any issues if I do as mentioned above when making maps.
19mm is in-between mine and the Ultimarc restrictor.