Are there any recipes for a Barnes* bullet and black powder? I doubt it. You wouldn’t be able to hunt with your ammo in the affected area.
*Barnes is the de facto sole source for “lead free” projectiles at the moment.
Rabbit hunters might want to look into 22 Hornet. It's just a little hotter than 22WMR and is a reloadable centerfire. Another alternative is 5.7X28mm. Centerfire and .224 caliber bullets. A Barnes TSX FB 45, 53, 62 and 70 gr .224 caliber bullets would do nicely. The 53 gr would be fine for .223. The 45 or 53 would be fine for 22 Hornet. I would go with the 45 on the 5.7X28.