This is naive. Your argument is hinged on the proposition that rich countries will remain rich and poor countries will stay poor. Given the current economic activity, a shit ton of countries will become rich and their resource consumption will increase. It's not if, it's when.

So stopping immigration will help,maybe, in the short term but the earth doesn't care for what happens in next 4-5 years. The earth does care if resource consumption increases exponentially over the next century.