It would be interesting to hear from anybody else who has asked their Granada hotel to obtain Alhambra tickets and whether the same request is made especially releasing the security number.
I can understand that you are making a commitment to purchase something and the hotel will have a system to ensure that when they purchase the tickets on your behalf then it is you who is paying for them up front. This is just like yourself paying directly using Ticketmaster web site. They ask you to input the same info.
Maybe the hotel in Granada (and other hotels) have a specific purchasing method which allows them to buy tickets outside the general public normal Ticketmaster route ? But they still need to have the purchase authorized and physically achieved rather than paying on behalf of the customer and then finding that the customer couldn't make it to the hotel after all.
I have many times used my CC for making purchases using web type forms and never felt uneasy but as you are clearly worried then perhaps you should telephone the hotel and discuss with them and then pass your detail directly by phone.
Personally I would just consider the standard of the hotel and whether I was confident they were people I am happy to stay with under their roof and if so would not mind passing my CC detail so I could visit the Alhambra.