Pick another example of a business that is a public accommodation that takes on clients:

A kosher deli that caters weddings must cater a Muslim wedding.
A maid service must clean the house of a satanist.
A welding shop must weld a statue of the devil.

Using your logic, denying these services discriminate on the basis of religion, which violates Title II of the Civil Rights Act.

Clearly this is not the intent of the law. And so, again, there is a material difference between creatively participating and an arm's length transaction.