However, it IS a religion and practice of it is protected by our First Amendment.
And that's what needs to change for Islam. as it's specifically designed to install a violent and oppressive government to overthrow that very document which you say protects it right now ... :-)
I won't pretend to know much about Islam. Not interested in it. However, you cannot change the First Amendment protections for practitioners of any religion. There are better ways to deal with the radical Islamists among us than to trash the Constitution.
For example, although I'm very wary of sedition laws and the misuse to which they could be put in the wrong hands, I would like to see the legal concept of sedition revived. If we had good sedition laws, people who misuse their positions as clergy to preach jihad could be charged with a crime. To repeat, this is just an example. There are many laws already on the books that provide law enforcement with tools to contain the seditious among us.