Would it surprise you to know that literally that exact same argument was used against Black people to resist integration, and indeed, the same Civil Rights Act of 1964 I linked? That it wasn’t because the person was Black, but because the business owner had a religious belief that was incompatible with service? That they shouldn’t be compelled to provide a service they disagreed with? It is as spurious then as it was now.
The reason your example is bad is because membership in the KKK is not a protected class, not because businesses are not required to provide equal access. Businesses are in fact barred from discriminating against protected classes and must provide them equal access (in general). Except, of course, if the Supreme Court likes the protected class in question less than they do the “free speech” of another class.
So, yes indeed, the government can and does force people to do what they want.
They will make an exception if you’re a Christian apparently, however.