What you are dealing with is an evolution of the meaning of words. Kant, in his essay on Perpetual Peace, refers to the Roman republic and rule mob, which he assimilated to tyranny. To him, that was the definition of democracy. Today, that would be called ochlocracy.
In his First Definitive Article, Kant gives the three cumulative criteria for what he considers republican:
- Freedom of the members of society
- Dependence of all on a common law (rule of law)
- Equality of the citizen.
If we look carefully, a republic could have any of the three mode of representation: autocratic, aristocratic or democratic -- but since in practice Kant rejects the third one, it is easy to understand that to him a republic would be either aristocratic or monarchic. An "aristocratic monarchy" would mean that the represenatives of the people would be selected to criteria of adequacy. It may be strange for us that a constitutional monarchy would qualify as a autocratic republic in Kantian terms, but it appeared perfectly logical.
On the other hand, it is still perfectly acceptable today to designate a political system as a aristocratic republic, i.e. the Republic of Venice of the Middle Ages; in Europe most republics were so (notably Switzerland), until well into the 19th century.
I would say that once this is clarified, his thought appears quite consistent.
Indeed, Napoleon I was Emperor of the French Republic and this was not particularly shocking at the time, since he was simply reconnecting to an old Roman tradition. But in Kantian terms, we can imagine that he would have called his regime a despotism, because there was no separation of powers any more.
Now to clarify your assumption, at the time of the foundation of the USA, the term democracy did not have always have an excellent connotation either. The Founders did not officially present themselves as "democrats" and the word democracy does not appear in the US constitution. It is only in the 1820s that the term lost its negative connotations and progressively acquired its modern meaning (essentially one man one vote). This change is reflected in Webster's definition. Indeed the Democratic Party was founded in 1828.
The US Constitution, when it was conceived, was thus republican. In the US and Europe, the question was still hotly debated whether all citizens (male) would be or not allowed to vote, possession of property being often a key criterion (race being another one). Hence saying that the phrase "we the people" is representative of "democratic" ideology is an ex-post reconstruction that uses a modern word in an inappropriate context.
