If you want to ditch the emperor, and this is mostly just thoughts on what I read from the Wikipedia article, why not make the system at least formally a triarchy.
So there is the Secretariat, the Chancellery, and the Department of State Affairs.
Head of the Department of State Affairs is the Chancellor.
Head of the Chancellery is the Chief Judge.
Head of the Secretariat is the Chief Secretary or even Prime Minister.
The titles are of course opted for change, but I`m just giving a general idea.
So the easiest one is the Secretariat. That is obviously the legislative body, which drafts the laws and puts them into practice. It is elected like a Parliament and the leader of the majority party/leader of the majority coalition is the nominal head. Thus you might call him the Prime Minister.
Then there is the Chancellor, who heads the Departments, not the Chancellery. He is what we might term the executive. The Chancellor is directly elected, and he guides the government in the day to day affairs.
Then there is the Chancellery, which is a mix between the supreme court and the Canadian senate, being an area for sober second thought. I kind of envision them as explicitly activist judges. They cannot pass laws, but they can alter existing ones (within limits) when they are brought to their attention. Thus they don’t review all bills that get passed, but anyone can challenge them in court, so in practice any moderately controversial bill will wind up in their hands. There they make rulings on what parts are legal, and subject to shifting, which parts are illegal, and which parts are fine. A contested bill can pass through the Chancellery mostly or entirely unchanged.
The Chancellor (the executive) selects his department heads from the members of the Secretariat (doesn’t have to be from the ruling party) and they are put forward for review in the Chancellery before their positions are approved.
The Chancellor (the executive) also needs to get his big foreign policy decisions approved by the Chancellery (war and treaties, for example).
There would be different election cycles for the different groups. Say election cycles of six years for the Chancellor themself and the Secretariat, but the election of the Secretariat happens three years into the term of the sitting Chancellor and the election of a new Chancellor happens three years into the term of the sitting Secretariat.
The Chancellery would be elected as well, but maybe on a less direct method. A limited selection of candidates maybe. So there would be higher age restrictions for the Chancellory positions, along with limitations to certain applicable fields (law, academics, etc… very professional based). They might also be selected on a more federal basis rather than population basis.
Anywhooo, that’s the idea I thought of.