>> There were no Republicans in 1804 <<
Wrong. Jefferson and Madison called their party “Republican.” But Andrew Jackson’s folks changed the name to “Democratic.”
And in between, the party was sometimes called the Democratic Republicans.
But Hamilton definitely was not with them. His party was the Federalists.
Fair enough to say that there were still no Republicans in 1804 in the modern sense. Jeffersonian Republicans were more dedicated to republicanism FWICS.