I don't have to make Islam illegal; sedition is already illegal.
Islam is a legal religion.
You don't have to prosecute them for adhering to Islam; you can prosecute them for practicing sedition.
An Islamist cannot even reliably repudiate Sharia law, because under Sharia law, all such promises to infidels are void. Thus, Islam is effectively an illegal religion, because its adherants automatically violate their oath of citizenship the moment they walk into a mosque and pray a prayer to impose Sharia law.
It has all the protections that apply to other religions, and as such cannot be excluded when other faiths are being included.
Who says they do? Get a clue: there is an authority here. There is nothing in the establishment clause that proscribes the power to prohibit a particular religious practice, just as there are laws already on the books against (even voluntary) human sacrifice that effectively prohibit certain pagan rites.
So you're comfortable with using legalities to outlaw a religion that you want eradicated.
Excuse me, but I'd rather we be honest if we're going to do that.
Of course certain practices of a religion may be illegal. Same for the Mormons, but outlawing multiple marriages didn't make belonging to the church itself illegal.