I did, because I had never heard this claim before. The main point--that the District of Columbia is foreign territory, and that it is controlled by London and the Vatican, is, to phrase it gently, simply incorrect. And I worked from your source.
There is much to analyze, but I am going to point to just one single thing: a truly terrible grammatical misunderstanding. That source you give quotes this sentence from the 1783 Treaty of Paris to "prove" that the treaty ignored the fact that America had won the war, by apparently stating that "Prince George" was the supreme potentate over a list of territories "and the United States of America."
No. That is an inexcusable misreading of the grammar. I have copied the relevant sentence below, with the important words bolded:
"It having pleased the Divine Providence to dispose the Hearts of the most Serene and most Potent Prince George the Third, by the Grace of God, King of Great Britain, France, and Ireland, Defender of the Faith, Duke of Brunswick and Lunebourg, Arch- Treasurer and Prince Elector of the Holy Roman Empire etc.. and of the United States of America. . ."
That is, this was a treaty between Prince (King) George III and the United States of America. And anybody who could misread a sentence so horribly cannot be counted on to give reliable information in other areas. That whole claim of "two Constitutions" is wrong to the point of being silly, and is not to be taken at all seriously.