3-week check-in on the hard joystick cap:
tl;dr: use a crumpled 3x3mm piece of paper and put it into the cap where the joystick is then inserted. Press the joystick against a hard surface to compress it but be careful not to press too hard or to let the joystick move.
At first I interpreted what you said to mean that I should put paper at the base of the joystick, which didn't help. I still had problems moving the joystick so I got the idea of looking for an o-ring. I've seen tiny ones before but didn't see one small enough in either of the closest hardware stores. I found an old computer part that was basically a small flat metal-like donut meant to go between screws and motherboards, but that didn't help.
At that point I realized that maybe you meant the paper should go inside the cap. My first try was too big and the cap stuck out too far. I had to use a pin through the top of the cap to push it out as I didn't have any other way to get it out (glue on the end of a push-pin occurred to me but wasn't available. Fortunately the pin didn't harm the top much.
My next attempt was a 7x7mm corner from a Starburst wrapper that I wrapped into a ball before feeding it into the cap. At first it was too big but once I turned off the power and pressed the joystick into the desk it compressed enough.
The joystick now works almost as well as I remember it working with the soft cap. Maybe it's just that I'm paying more attention to it now, but it seems like the mouse sometimes doesn't move on the first attempt and it's easier to move left/right/up/down compared to diagonal. Again, this might have been the case before and I just hadn't noticed. My last soft cap wore out so none of my Twiddlers have one I can use for comparison.
Edit: I used scissors to remove the 'brim' of the cap and now the mouse movements are fine in all directions. Without the stuffing method above the brim would prevent mouse movement as the joystick had no room to 'tilt' in the desired direction. With the stuffing the brim would hit the rim of the joystick hole, most notably in the up and up & right diagonal directions. Removing the brim means the cap works flawlessly without stuffing, but unfortunately it leaves this Twiddler open to dust and dirt. Hopefully you can tweak the hard cap design a little.