There is a paper thin enough to patch a .224 boolit that might fit a fired neck without sizing either boolit or neck and that is baking paper (or vellum). Tough stuff. It measures 0.05mm/.002" so a double wrap would increase the diameter to .232 which is fine if not a little small. A thicker paper or three wraps would bring the diameter up to seat firmly in an unsized neck. Seat just deep enough so that the boolit touches the leade without the base protruding into the case.
Another option is to make bullets using spent 22lr cases. I have done this but it is a bit of a mission. I soldered into the heated case then swage them down - the excess solder extruded out a bleed hole in the nose of the die. I never did try them (only made two).
You might even find someone to make you a push through sizer. I would offer but I know it takes me forever to get round to doing things.
I have patched a few boolits for my hornet.
https://i.postimg.cc/dtX4tfdW/MVC-047F.jpg
The paper was a thick tracing paper and no sizing was done.
https://i.postimg.cc/Prc8wsSw/MVC-050F.jpg
I can't tell you the accuracy but they worked fine into a 'test tube'. These were .224 cast, wrapped with thick paper fired in a .223 grooved bore.
So I would expect a .224 boolit patched with ordinary paper to work unsized.