4 point sockets are out there, often they are called "female pipe plug sockets". I have only ever seen them down to 1/4" square though. I've also never seen nut drivers that small in 4pt.
One option you might want to look into would be clock winding keys, or clock mainspring let down tools. You can find clock winding keys with 1/8" square female ends. They might alternately be labeled as a #3 (3mm) or #4 (3.25mm) keys (1/8" is 3.175mm). Something like that might work for you, but many of those tools have quite a bit of chamfer on the leading edge because they are designed for easy engagement with a square shaft instead of a nut, so you would want to either grind that chamfer off yourself or find one with minimal chamfer so you could have good engagement when used to turn a square nut.
Examples of what I'm talking about-