It's a near five hour drive from Alicante, but the Jarapalo club near Torremolinos is both gorgeous and outstanding. (Google Tiro de Plato Jarapalo and have a looksee)
While I was there I booked a day of instructions after talking to Pepe Sanchez (I asked around, rang them up and was invited over straight away) and he brought a gun which he adusted on the spot to my needs. Had a great day and wasn't too pricey either. Next time round I'll bring my own gun (and some fellow shooters) and make arrangements accordingly. Pepe talks to the customs/police people at the airport beforehand if you ask him too, which saves you having to answer too many questions in Spanish, as they'll know who you are, what you are bringing and why. I have a European pass so should be fine either way, but expediting the process of clearing customs is preferable. They also let you store the gun in lockers at the club while you are there, I believe- but do check their opening hours in order to pick up your gun prior to depature. I got a lesson there on a day the place was officially closed, but because of it Pepe was able to help two Brits to their guns and send them on their merry way. Otherwise, either no plane or no gun...