We have that in the US already, with the evangelists trying to do the exact same thing - strict religious laws and anti-homosexuality. Lots of gays/transgendered people being killed in the US, as well as domestic violence causing death among people that stray away from their family. And what good are laws against murder if the government doesn't enforce it, like the current US Attorney General is doing?
But, it doesn't matter since immigration isn't designed to influence culture. The US government isn't a Cultural Government. We don't even have a national language. Pretty cool, huh?
Immigrations instead is used to maintain economic welfare, as immigrants work harder than natives and have a lower crime rate. Otherwise, you have a stagnant economy like Japan.
You have plenty of other options to control culture besides immigration. Learn how to make a TV show, or speak in public.
So, the correct answer to everything is to make sure we have an open border. Let's work to make sure that happens.