I got the screws from McMaster today. Instead of ordering the 8-32 3/16" long, after measuring several switches I found that I could order the 8-32 1/4" instead. Even the 1/4" still has plenty of room left inside the switch. The only reason I ordered the 1/4" instead of the 3/16", is that the 1/4" comes with a 90 deg cone, which is slightly more "pointy" than the 110 deg cone in the 3/16" screw. Not anything really earth shattering, but since both fit fine inside the switch, I went with the more pointy screw.
Here is the new screw (right) next to the original OEM part (left):
Close-up, where you can see that the new screw (even at 90 deg) is almost blunt compared to the OEM screw. Here you can see why the advice from
LuxLuthor earlier in this thread makes sense :
The small bag of screws that I bought:
And since I really like the screwdriver handle that I ground for the T8 screw, I got the same basic handle for the 5/64" screws: