Pay a thousand bucks to a Montana agent to register you an pseudoanonymous LLC and put an old 90s Corolla into an LLC with a permanent registration plate (since anything 15+ years old can have a 'perm' plate on it.

Then never think about it again.

While some people do get away with this, it carries some risk.

Without using an LLC, most every state requires you to register your vehicle where you live within 30-90 days with some exceptions (ie college students).

Even with the LLC, if you catch the attention of the state, I believe you might be risking being charged with tax evasion even if your goal was to protect your privacy. This is especially true if you can't prove the LLC to be a legitimate business venture.

Yeah, the Corolla won't be mistaken for a supercar, but many states have begun cracking down on residents with Montana plates such as Georgia, Ohio, and New York.

Also, insuring a car with out of state registration can be committing insurance fraud. Rates and fees are different between states due to different regulations. Further, depending on your policy, the insurer could deny claims because the car wasn't garaged in the state it was registered.

Really, if the privacy is of sufficient priority, the best solution is to just do things properly and move to rural Montana instead.