I can vouch for the quality of Four Corners transport as well- used them several times, and I've recommended them to family as well.
However, in this case, If I were you I'd just pick up a Harbor Freight trailer and use that. It's perfect for your situation. Get the one with the bigger wheels, and mount a wheel chock to it to secure the bike. A friend of mine did this from GA to NM, then NM to AZ last year. Still has the trailer. It's small and light enough you can tow it behind an ordinary car and won't even know it's there, and it's cheap enough that you can even afford to keep it after you are done moving.