It's been some years since I did that project but going from memory, I just took the socket, with a QR sling mount in it, to Home Depot or Lowe's and found a wood screw that would still allow the sling piece to still pivot (I don't recall which type).
Drill a pilot hole using the bit for the screw then drill the hole for the socket. Put just a little epoxy in the screw pilot hole (not too much as you don't want it coming up through the screw hole in the mount), wipe a little inside the socket hole walls then stick something in the screw hole in the socket (so epoxy doesn't get squeezed into it) and push the socket into place. Pull the stick out and screw in your wood screw making sure you don't get epoxy inside the swivel cup. If you do, some acetone on a Q tip should get it out. Allow to cure and you should be able to hang from it without it popping out. Alternately you could inlet the stock for one of the other QR type mounts but that's more work; with the wood screw and epoxy I don't think the single screw cup style it's going anywhere.
<--- Not a gunsmith or machinist and my woodworking is more like a beaver that needs braces but I do enjoy tinkering with inexpensive guns.