Where is it written in the Constitution that if a religion has political aspects it ceases to be a religion?
Mormonism in its early years was a totalitarian religious/political system every bit as much as Islam was/is. The medieval Catholic Church was also a monolithic religious/cultural/political system. No dissent allowed. Were neither 19th century Mormonism nor medieval Catholicism religions?
The BIG problem with your idea is that it would require giving a government agency the power to decide which groups are and are not religions. This is a BAD IDEA.
“This is a BAD IDEA.”
Got a better idea to deal with Islam?