Well, I solved it for $1.75 and a few minutes with a hacksaw. I unthreaded the hex shank from the chuck, and replaced it with a 2" long 3/8" x 24 fine bolt, with the head hacked off. Fits perfectly and is perfectly snug. Thanks all for your ideas!
According to an ebay seller, the “G2” sockets are shallow, not fully broached.
Until I can get a positive ID on a socket, I‘ll look into alternatives. A slightly wider shank (or just a bolt) on the chuck seems like a great option.
Ok, I've been searching, and have yet to find what I'm looking for. I need a fully broached 1/4" deep socket. Shouldn't be that hard. But so far I've struck out. Do you have a brand where you can confirm the 1/4" x 1/4" is broached to 1" deep or more? Don't care if it's chrome or black. I'd...