Some may say I'm a fool for doing it this way, but when we relocated from CT to TN last fall I used one company, ATLAS VAN LINES to move everything for one firm price. They charged for the weight of the entire load by the pound. The load included all our furniture, household belongings, lawn mower, ect AND the bikes - everything you take with you when you empty the house. Just make sure the local agent for them has a good reputation by checking reviews and asking people who have used them because the ATLAS agents are independently owned and operated and therefore are not all created equal...
The truck driver (who was also an independent contractor working under contract to the ATLAS agent) and I worked together quite well when loading the 3 bikes ensuring we were both satisfied that everything was secure - I did have an issue with the moving companies dispatcher and a few items were temporarily misplaced....

.. a long story for another time and place.... But the important thing is all three bikes got here without any problems. And in the end everything we packed and inventoried found it's way here.
FWIW they were the only moving company I could find that would transport everything in one truck and not subcontract out the vehicle transport for an additional (and astronomically high) cost.