I have a friend who works the counter at the USPS. Express Mail is the preferred way to send live birds. If you live near major airports, the service is guaranteed overnight. If not, it can be two days. I am in the NYC area, so I don't have problems receiving overnight packages, as long as they are coming from towns that are also overnight areas. Many far out rural areas are two day express mail zones.
Express mail is always given special handling over priority mail. Express should always arrive faster than priority because it is received by the post offices more than once a day and is not part of the regular mail delivery service.
When I had eggs shipped priority, they arrived in two days. When I had day old geese shipped Express, they were here overnight. You should take the zip code to which you are shipping and ask your postal clerks to check the shipping times for express and priority. Their computers should be able to tell you what is the fastest method. I swear that I read somewhere that live birds HAD to go express mail because of postal regulations, but I could be having a Senior Moment and my mind could be fabricating information on that one.
Good luck!