It is entirely legal as both of these species are not-native to the USA.
Is there a law that makes it explicitly legal to make traps for non-native species? Or is it just not forbidden?
I know some species can do serious harm when introduced in a new area, ask Australia. But on the other hand that is how nature works, all species come and go (we too)...
Some questions arise
Q: is your trap lethal or non-lethal?
Q: How do you prevent that only the "non-native" birds are trapped?
After reading your post I couldn't help thinking the following:
Most Americans are not-native to the USA. They originated from Africa, spread to most of Eurasia 70000 years ago, and migrated to America only a few 100 years ago....