I don't think you should have too much difficulty finding space at any of the three naturist sites mentioned.
We haven't ever been asked for INF cards in Spain to my knowledge, however, the website does say they're mandatory at El Templo del Sol - we stayed there about four years ago.
Membership of British Naturism, the official national organisation in the UK, includes INF Membership - the BN card is endorsed with the INF logo and that's it.
I expect you can buy membership of INF at El Templo, but it won't include membership of BN, which I think would probably be better value. In fact, purchased at the gate, I suspect it may well include membership of the Spanish Naturist Federation - but I don't know that?
You could just wing it and see how you get on, or join BN pronto - best to speak to them on the phone tomorrow, if you want to go down that route, in order to see if they can get the cards to you in time.
Final thought about El Templo, if you encounter a problem, there is a textile campsite a few hundred yards away, that would enable you to walk to El Torn naturist beach - not the same as staying at El Templo, but another option.
Whatever you decide, good luck and have a great trip. Let us know how you get on.