Just returned from an ibex hunt there yesterday. Took my rifle. I did use Travelwithguns.com for assistance on flights, and they will inform you what permits you need and what the process is, but then it's up to you to complete the process with the Spanish Consulate that has jurisdiction in your area of residence. This turns out to be important, as believe it or not, different Consulates can have slightly different requirements/forms, and they will not accept forms from the other Consulates.
I considered both bringing my rifle and using the outfitter's rifle. The permit process is about the same either way as you need an invitation letter from the outfitter (which will include the make, model and serial number of the gun you will hunt with whether yours or theirs), and with that you can start the application. Since the application is for permission to hunt with guns in Spain, it doesn't matter if yours or theirs - you will go through the same permitting process. My gun is a 6.5 PRC, the one he had available for me was a fine Merkel 6.5 Swede. My shots were at 300 yards and 80 yards on Gredos and Beceite ibex, so it would not have made much difference. I could have made much longer shots on both, but stalking in closer was very doable in both situations, so we did.
At the end of my hunt, they asked me to take a management ibex in an area they have been asked to reduce the population. That one was at 630 yards, and I was very glad I had my gun.
The hunt was one of my favorite of all time. Great outfitter, guides, food, accommodations, and game populations. The hunting culture there is strong and important, and you feel welcomed. I will definitely be going back and highly recommend the whole experience. Enjoy your trip!