There is a difference between spreading "democracy" and spreading representaive government.
And Hamilton, as commander of the US Armed Forces during the Quasi-War, did suggest seizing both the Louisiana Territory and Mexico by force - implying that he saw Mexico as potentially being absorbed into the US or being organized as an ally along the lines of the USA.
+1
Hamilton was an old fashioned imperialist (e.g. wanted the U.S. to conquer territory like other “great nations”) but that has nothing to do with the Modern Wilsonian, and neo-con, concept of spreading democracy. The primary motivation for Hamilton was old fashioned power politics not bringing “’representative government” to the world.