Gotta do more pictures, I guess. Lots of words in those pixels.
Any device built for a single-purpose operation on the lathe should be designed around the most secure and easy mount available. That's why even using a premade sanding disk made for a tablesaw shouldn't rely on grabbing some bolt in a chuck or sleeving a tapered buff mount, rather create a mount to which the disk is bolted which can be held in the chuck, or use a dedicated (forever) faceplate.
Same for a disk made to order in size. Especially important to true the face on such a disk, which might cut through the hardened and compacted surface of MDF, exposing the more open inner structure. This will be more difficult to get glue to adhere to than the slick outer. Shellac provides a good way of getting a quick, slick surface suitable for glue. For me this means sticky back paper or spray mount like 3M 77. Hot glue would not be my choice, because it can leave lumps.
Finally, the dovetailed block on any device should be sized so that the largest jaws you use are a close match in size. This will ensure that the circularity from turning it originally the lathe is what you will get when you mount it over and over. The load from the metal jaws will be divided over a broad area of the softer fiberboard, leaving no marks. Minor defects caused by rough handling can be taken below the local surface by sanding without affecting the hold, because they will be bridged easily by smooth jaws which reference themselves to the average surface. Of course, they will also reference themselves to the rear of the disk as they are tightened, wedging up by their design. A tenon is smaller and easier than using a large stabilizing block with a mortise. Another plus on something that will be set aside often is that it won't get dust and shavings inside which can hide and cause problems with the inside hold. Remember I said that a dovetail was a no-compromise hold? If it doesn't mount perfectly, it's because something's in the way, so look for it.
Oh yes, there's a second measure of safety in a dovetail, because loosening chucks won't throw the piece without warning. It'll just start to wobble.
That's principles, final design based on them should be a success.