Can I take down an invasive bush with a possibly abandoned robin nest in it?

There is a bush in my backyard right up against the house that has been a pain for at least a decade. I believe it is an invasive species (which I don’t want near the native meadow I’m building), and has been slowly pulling off the shingles at the corner of our house. For a couple years, there have been robins (Turdus migratorius) inhabiting a nest in the bush, but from what I can see, there have been no offspring produced. I believe the nest has been abandoned again this year. I want to relocate the nest into a nearby bush that we plan to keep, and chop up the invasive one so it dies and I can hopefully pull it out from the shingles safely. Would it be legal to complete this nest relocation right now, or should I wait til the off-season when the nest will definitely be vacant?

Edit: I live in the Northeastern United States