Would anyone have any suggestions how to make this trip? We are willing to rent a car but couldn't find anything reasonable so far since we would want to leave the car in Bayonne, so were trying to think about other ways to do this. We would be leaving Barcelona on June 24 and have to arrive in Bayonne on June 26 for Rick Steves Basque tour. We did think maybe we could do a Barcelona to Pyrenees tour and leave the tour in the Pyrenees area, spend the night, and then somehow travel on to Bayonne the next day.
24 Jun Barcelona - Puigcerda (3.25 hrs)
http://www.trenscat.com/renfe/r3_ct.html
25 Jun La Tour de Carol - Toulouse (3.5 hrs)
http://www.ter.sncf.com/midi-pyrenees
26 Jun Toulouse - Bayonne (3.5 hrs)
http://www.voyages-sncf.com/billet-train/horaires
The simplest solution would probably be to take a train from Barcelona to Perpignan, France, and pick up the car there. Then, you can drop it in Bayonne without a 2nd-country drop fee.
If you were to leave the car on the Spanish side of the border, such as at San Sebastian (aka Donostia), then it takes 2 trains to get to Bayonne; see this recent thread.