A tangential thought....
If you buy one of the cheap hole saws, first use it to make a hole in a piece of MDF to act as a guide later.
Grind off the teeth, and sharpen the edge, remove the drill bit and use the MDF as a guide for the new cutter.
Cheap ones tend to be made of Plasticine, so the edge probably wouldn't be long lasting, however IF it works you could consider upgrading the tool to a better quality one, or it may be just enough for limited cuts.