creating something like an expanding ring mandrel for my homemade lathe
Making one properly without a (metalworking) lathe would be very difficult.  With one it's quite easy.

This youtube video shows one made from simple hardware, and you could choose an appropriate screw for the shaft to fit your chuck.  But I'm not sure I'd trust its centration (might not matter if you're just using it like the second video link).

It may be possible to replace the chuck on your drill with a 1/2" version.  Then you could buy and off-the-shelf ring mandrel.  1/3" is uncommon, though 3/8" or 10&nbsp;mm is quite common on battery drills, and I have a hand drill with a 1/4" chuck.