I would not recommend messing with the Constitution in an effort to address Islam and the obvious contrary ideology from that expressed in our founding documents. Simple recognition of their law being opposite to our own should suffice. I personally refuse to accept them as a religion. Our founding elements were familiar with Christianity and what it was capable of in the hands of fallible man despite principles or commandments to the contrary. To consider any other comparison to prove a religion over criminal enterprise, or false ideology, would not be true to our founding IMHO.
That said, they would probably want to take it to the Suprmee Court. That might be interesting.
The Supreme Court cannot overturn a Constitutional Amendment even when it is in conflict with another. They can interpret, that’s all. So, a simple amendment not imposing anything other than it is not a religion is what is needed. Once is is not a religion, everything else falls into place.