You said it yourself. Democracy is merely a tool for the US. They do not care about democracy, they care about their hegemonic ambitions and nothing else. If promoting democracy helps achieve that ambition (even if the new puppet doesn't turn out to be a democracy), they'll use it.
Though I still refute what you said, the US doesn't even need democracy to influence. It has economic & military incentives, as well as coercion (using the same themes). In my mind democracy is just a "buzzword" for them to use to get world opinion on their side.
They don't need it in countries that are already willing to submit to American hegemony.
But they'll promote it in places like Iran and Syria because those countries have so far refused to play second fiddle to them.
If Iran suddenly decides to submit to American hegemony, then the two countries will become friends overnight and all pro-democracy rhetoric towards Iran will quietly disappear.
Still, however, democracies are easier to manipulate than dictatorships. If the US gets into a future disagreement with its new Iranian ally, then it won't be able to peacefully remove the Iranian leaders from power. But should Iran be a democracy, then the US can always make sure it lets pro-US political parties win the general/presidential elections.
So, in the end, America will by default favor democracy to other forms of government.
The real question is whether becoming part of the US sphere of influence is necessarily a bad thing. For countries like Germany and Japan, it has worked wonders. In exchange for becoming part of the US-led international system, these countries have benefited a lot in terms of economics, technology and science. Moreover, the citizens of these countries enjoy many kinds of freedoms and liberties.
So is it that bad? The world has always been that way.
Finally, becoming part of a US-led international system does not necessarily mean Iran will lose its autonomy. At the end of the day, all countries are autonomous and independent decision-makers, including those that appear to act as puppets to greater powers.