You may not have received any replies as this is an extremely long request at just under 1100 nm. Long distance transports are possible but require more coordination than short transports. Private planes do not normally fly across country like commercial jets. Private pilots usually only fly about 300 nautical miles and should not be expected to land and refuel and continue on. There are exceptions, but this is a guideline. Long transports need to be broken down into legs of about 300 miles each. This requires pilots to coordinate their schedules and meet at airports to transport the pet on the next leg. The longer the flight, the more connections needed. You must allow for contingencies such as weather delays or mechanical problems, so you should arrange temporary foster for the animal at each connecting point in case of an unexpected layover. Ground transport or commercial air transport will be better alternatives for many long distance transports.
You can contact pilots directly. Pilots indicate the best way to contact them in their profile, commonly by email or PM (private message). Click the pilot's profile and look for an email icon or a PM icon. You can also see these icons under the pilot's name on posts on the Ride Board. Some pilots allow other forms of contact, and if so, those will be indicated in their profile, signature line, or the body of a message.
You can also find much information here faq.php?mode=rescue
You might also want to look into a commercial flight or Operation Roger--the OTR truckers group.