So why is the opinion of Lee’s cousin as to the ‘correct side’ of greater significance to the opinion of Lee himself?
Because you said, "You are assuming that the Union = the United States, which, in 1861, was not the case."
You were contending that an agreement existed about federal versus state citizenship -- that one was a citizen of a state, rather than a citizen of the country.
There was no such agreed upon understanding that one was a citizen of a state, rather than a citizen of the country.
It may have been convenient for people to convince themselves of that, but it wasn't the case, particularly for people who had taken an oath to the federal constitution and laws.