This is a flat, substabtial interface with the 10th Amendment’s allowance of states to manage and run elections. If a state allows elections such that they aren’t denied on account of race, gender, and people under 18, they can be run however the states deem. Requiring someone to register with the federal government is not a power of Congress
What are the odds of blue states setting up their own mail services in response to this?
Like near zero, no one understands how expensive delivering mail is and how much of USPS funding is tied into real estate. If USPS was restricted from shipping packages they would go under tomorrow.
I doubt it can be done in time for the election.