If you have a drill press, you could get a 2 inch Forstner bit to cut the circle, or a hole saw might do if you drill the top side halfway through, then turn it over and cut the other side. Use a roundove bit to make a graceful looking mount and remove any chipout. Then cut using a stop block to make certain both pieces are equal length.
Instead of making a flat bottom, consider using dowels to connect the two ends. Drill the holes on a drill press using a stop block to position the holes precisely. If the wood is beautiful use a Cherry stain, or if you want it to look instrument-ish, consider using a deep mahogany (rich-dark red) stain and then wipe on poly. If the dowels are too tight, lightly sand the ends to slightly reduce the diameter and glue. Let it dry in a jig so everything stays flat and aligned. This will greatly speed up the project, make the stands very light weight, and it will have a nice look. So quick to make you could offer them with the kaleidoscope if you're selling them. Two dowels set vertically will look nicer than the flat bottom and be at least as strong.
Actually, this would be a great item for sale at a craft fair. Get an ordinary kaleidoscope, put a new exterior wrap on it wood patterned paper or even veneer, paint the metal rima color that matches the wrap or the stand (unifies what then becomes a set).
If you don't like the dowels, I would substitute a vertical piece with rounded over edges that spans the two ends and use a router to cut a groove for it. That will require a jig to make a stopped grove with round ends. Personally, I'd prefer something like a 3/8ths to 1/2 inch pair of dowels, about the same thickness as the end pieces. A single dowel would work if you had an assembly jig to asure it was perfectly aligned.
If you're going to go with a flat bottom piece, use a thicker piece the width of the vertical pieces, then cut a groove across the full width, but add a flat piece of scrap where the bit will exit to prevent blowout. If you don't presently have a sled for cutting exact fit grooves or dados, consider making one like the picture.
The more I do, the less I accomplish.