I've never heard of any travel restrictions for cockatoos....
Only travel restrictions I've heard about have been with quakers, indian ringnecks, golden conures and maybe nandays - dependent on the state.
Many amazons, military macaws, scarlets, red front macaws and others are also on Appendix 1, and I've never heard before of any of these common (in captivity) species having travel restrictions put on them, whether or not it can be proven where these birds were bred. The only thing I can find is in regards to travelling internationally...
Anyway, for a blue crown conure, it shouldn't matter whether or not the bird is banded! The majority of my birds have not had bands, including many that came from pet stores. My cherry head didn't have a band but my mitred did, and it was because of his band that I was able to find out when he was born (well, what year), that his parents were wild caught imports (so he's a first generation captive bred!), and the general location he was born at (he never moved far from where he was born until I adopted him). I honestly wish I could have found out the same information, or more, via my cherry head.