I'm not sure how people would have felt about a force user being in charge of the Republic. Obviously that happened in Canon anyway but only few knew about Palpatine being force sensitive. Dooku, on the other hand, was a Jedi for a considerable part of his life. And, from what I know, he also seemed to be relatively well known. So, it's quite likely the people of the Republic would have known about it. Which would obviously raise questions about his affiliation to any fractions of force users, namely the Jedi Order. I don't think people would have felt comfortable with the thought that politics could be steered by a bunch of people with super powers. After all, the Jedi didn't have all that much political power in Canon, either; they served more as some kind of police force as Lucas once described it, I think, rather than an actual political party.

So, not sure, how that would influence TCW overall, but I could imagine that most senators would be less trusting towards Dooku as the Supreme Chancellor and, as a result, less inclined to grant him emergency powers

Dooku would have been bogged down by the politics of the republic and the relationship between them and force users and based on his reputation it seems that count dooku was well known enough to garner attention