You certainly should not NEED a JAA licence - as the privilidges afforded by ICAO say you should be allowed to fly privately in any ICAO member state.
However, it's up to the Spanish Authorities as to what hoops you will need to jump through (if any) in order to rent a Spanish registered aircraft and fly it in Spanish airspace.
The opinions of anyone on this board are academic - it's only the Spanish equivalent of the CAA that can answer your question.
Google and e-mail time, me thinks.