If this is chrome-lined and has a chrome-lined chamber I would not re-cut the chamber. It could get messy. Hand load for it exclusively, shoot .223, or swap out the bbl.
I personally don't think it's a problem shooting 5.56 in a .223 from experience but I would not give that advice out loud on the internet.
It sounds to me like you'd benefit shooting heavy for caliber .223 varmint loads anyhow, given your intended use.
Pull some pipe cleaners soaked in chlorine-free brake cleaner through that firing-pin channel.