Swaging the rim thinner could be an option. I used to make .455 cases from 45Colt and used the method to 'thin the rim.
You do need to make or have made a simple die that the 22H in this instance will slip fit into and the rim land solidly on the bottom flat of the die. The interior of the die does not need to be chambered formed to the Hornet. Only the base dia of the case need be a close fit to the die.

The .455/45Colt swage die reduced the rims by around .025. I should think it would work as well for
the Hornet too if it s need around .010 reduction for the conversion.

The easiest way IMO is if the die fits into your reloading press. Then using a small primer pocket swage in this case, push the 22 hornet case into the die and adjust the die so that the case rim is 'squashed' against the base of the die.
Use the top of the linkage toggle movement where the leverage is the greatest.
You adj the die in and out of the press as well to make changes in the thickness of the rim.
The Primer Pocket Swage works well as it not only provides a flat surface at the base to push the case into the die, the primer pocket is supported during the process so there's no chance it can close up any.


I've seen others make a similar die but with a rim recess on the bottom of the exact depth they want the case to come out. Then the die with unaltered case inside is placed base down on a very sturdy steel surface.
The die then struck w/a hammer.
Kind of like using the old 2 piece bullet hammer swage idea.


One small thing you usually have to do is trim the outside dia of the rim back down. The brass you are moving has to go somewhere and it moves to the outside dia of the rim. You can spin those in the Black&Decker and trim w/a file if you
don't want to buy a lathe.
....Maybe you could have the other end of the same die cut with a rim recess of the correct dia and use that end with the same primer pocket swage to push the thinned but too large in dia rims into the shear the excess off.
Just a thought
Then there's that pesky rim chamfer,,back to the drill and files.

Good luck with your project.
I have a rifle that was/is chambered in 22 Vierling, Originally a single shot that looks like a undersize Mannlicher Schoenauer.
But someone made it into a repeater,,quite nicely though. I've never gotten around to checking that chamber to see if they altered it or not