You could use the BA lounge as an alternative, it looks to be open from 5:30 am.
Having said that, I've tried both the BA and Plaza Premium lounges when flying out of FCO on BA. They weren't really much different, though the BA lounge does not have bathrooms. There's limited à la carte options in the BA lounge, but the buffet in Plaza Premium seemed more then adequate for breakfast. Plaza Premium has barista made coffee.
Assuming the same level of crowd, I'd probably opt for Plaza Premium over BA's offering.