Two facts about me well known here:
I am seriously deficient in the ingenuities department. Meaning, if I can visualize and complete this project it is kiddy-garden simple.
And, I ain't gonna pay no $100.00 for something when I can make it for $5.00.
The cutter you see was part of a package of 10 I purchased from Grizzly for about $25.00.
The 3/8" rod cost a buck or two at the local steel supply. The wood came from my scrap pile. The ferrule was probably the most expensive part, a coupling from the hardware store, almost $5.00. Never do that again.
The carbide cutters are available from many sources. They are also available in round. But, for whatever reason, round ones are a lot more expensive. They range about $10.00 to $15.00 each. Don't ask me why.
For your project, I would recommend you get a round cutter.
The shaft can be almost anything but square stock in 3/8", or larger is best as it lays steady better than round.
For what you are doing, a long shaft with the handle end bent at 90 degrees would give you a good grip. But, please, do use a rest.
Don't use cold rolled steel, way to hard to work with and is not necessary. Hot rolled is much-much easier to work with.
Grind down a flat, drill a hole, tap and screw on the cutter. BTW, buy your screws with the cutters.
BTW, the carbide stays sharp a very long time. With square you can just change edges. With round, just turn a little. One cutter will last and last and last.