Anything that you come up with will need to mount to the forward fixed barrel portion.
The weight of the slide on these guns is finely tuned to work along with the recoil spring. Stick a rail and red dot onto the moving portion and it will add enough weight that the gun won't cycle correctly.
I know there are special very lightweight open frame style red dots for handguns where they replace the rear sight. But they are generally used on center fire pistols where the added mass of the site won't affect them as much. On a rimfire even a lightweight option like that would likely mess up the cycling.
There's two ways to adapt something if you're handy with tools. One is to cove out the base of a picatinny rail and then drill and tap the fixed forward portion of the receiver to accept some mounting screws. This is touchy stuff since you're drilling and tapping into the metal around the barrel. Not a good idea if you're not a keen and experienced metal worker.
The other option is something I recently saw done on a S&W 422. It was a replacement right side grip made from aluminium that extended up and "L'd" over the slide. The top of that L was a picatinny rail section that mounted a red dot. Now nothing says that a red dot has to sit upright. If you did this with a flat plate of metal and screw mount a rail to the flat side you could mount the red dot sideways. Just the adjustments would reverse. You may need to bend or otherwise build up the system to allow for any offseting needed to center the tube over the barrel. If done neatly it would not look any worse than any other .22 with a red dot. Of course this means once again that you need to be fairly handy with doing metal work to build such a thing. But at least in this case you're not doing anything to modify the gun itself. If you're handy enough to be able to line it all out, cut things with a hacksaw, shape them using files and sandpaper and drill/tap/countersink for screws to hold it all together then you could make your own mount.
Really though, it's a lot of trouble unless you're simply in a situation where your eyesight and corrective glasses makes it tough to use the plain iron sights.