Have you considered 5.6x50R? I have a Martini Cadet in that calibre and it works very well. I don't know about the US but here in Oz I have had no difficulty in getting brass (RWS for example - pricy but excellent), or for that matter factory loads (S&B).
You can also use 5.6x50R brass to make .222 Rimmed, or rimmed versions of other calibres based on the .222 head size.
