High-level: Use Existing Payment Systems
I think that this approach -- using credit card numbers to determine if a user already has taken advantage of a free trial and should be ineligible for a subsequent free trial -- is misguided. Firstly, you will drive away potential customers by requiring a credit card upfront (which many users don't give out unless they are actually ready to buy), instead of requiring it only after the trial period has ended.
Secondly, you are reinventing the wheel. There are a plethora of "app stores" (the Chrome webstore, the Android marketplace, the iTunes app store, etc.) which provide builtin mechanisms for payment and trial periods. Using these systems will give your product increased visiblity to consumers, will offer your potential customers multiple different payment methods (making them more inclined to buy), and will also save you the hassle of implementing this mechanism yourself. Plus, users generally prefer to give out their credit card to the least number of companies possible; not only would you have to implement this complex mechanism yourself, but you would also have to get users to trust you enough to use it.
Lower-level: Implementation Details
Any hash mechanism can have collisions, hence you would still need to deal with this problem. You should obviously use full disk encryption and other best security practices with your servers. The risk of having both the database and the salting algorithm compromised at the same time can be reduced by hosting the backend database service on a separate machine from the one that hosts this code. The main vulnerability of hashing is brute force attacks. And the best way to deal with them is to just make brute forcing expensive enough that it isn't worth the attacker's while. Using a separate salt for each entry (e.g. the customer's name, the customer's zip code, etc. as part of the salt) will make using rainbow tables ineffective. Of course making the data, itself, less valuable to attackers (e.g. not including the full credit card number) is also a good way to discourage these kinds of attacks. In any case, I again advise you to take advantage of the many app stores instead of implementing this yourself.