I would have to say that I do not see how Burr's acts could possibly constitute treason. They were certainly against the interests of the United States, but that does not rise to the Constitutional definition of treason.
The Constitution specifies that
Treason against the United States, shall consist only in levying War against them, or in adhering to their Enemies, giving them Aid and Comfort.
If this is the definition of treason, how could Burr have committed treason? His conspiracy, it seems to me, was meant to capture land from Spain. I do not think that he was actually trying to break off any part of the United States to form his "empire."
In other, words, I think he was filibustering, but I do not think that he was engaged in war against the United States.