Let me help delete the credit card information on your customer's profile.
As you've noticed, there isn't an option to simply delete or remove the credit card information on your customer's profile. Instead, you'll need to create an additional customer profile and merge it to the one with credit card information. By doing this, the new customer profile will have overtaken the old. Let me guide you through the process.
Here's how to create an additional customer profile:
Click Sales from the left menu, then select Customers.
Select New customer.
Enter the customer name slightly different than the original.
Fill in the other details as needed.
Then, let's now merge the two customer profiles. Follow the steps below:
Go back to the Customer page.
Select the original customer profile (the one with the credit card info on it). Then, select Edit.
Change the Title, First name, Middle name, Last name, Suffix, and Display name so they're identical to the new customer profile you just created.
Click Yes to confirm the merge.
Once done, go ahead and take a peek at the profile to make sure the credit card information is gone. Also, you don't need to worry as the transactions that were related to the customer profile with the credit card information will be merged and appear on the new profile.