Ancient Roman Senate Question?

All official positions were temporary (usually for one year terms): e.g. Consul, Praetor, Aedile, Tribune, etc. Priesthoods sometimes went with these offices... but there were some important priesthoods that were for lifetime terms. the Flamen Dialis was one... but by the late Republic, this priesthood was considered so archaic that no-one really wanted to hold this position any more. Pontifex Maximus, the head of the most important religious "college," was another. If you want something secular, with no particular religious angle to it, you might consider making your character Princeps Senatus ( the "First Man of the Senate"). This was not an official position: i.e., you didn't run and get elected to this office. Instead, it was a position of informal authority/dignity/respect. It was held by the oldest member or the one who had served most often in the highest offices. For instance, Cato the Elder, one of the greatest leaders of the Middle Republic period, was Princeps Senatus for many years. That meant he was given the right to speak first on any issue under debate. This position was later transformed into the position that we know as Emperor, but that didn't come until after the collapse of the Republic, under the first Emperor, Augustus.

The Princeps Senatus comes closest to what you want. But it wouldn't work in your story, because really secret organizations didn't exist. That comes much, much later.

During the Roman republic and long after that politics were individual. Parties as we know them didn't exist. Yes, there were conservatives ('Boni') and progressives ('Populares') but that weren't real parties. Every politician was in real serious competition with all the others. A politician would normally be one or the other, but that was not permanent. In order to score points it was completely normal to change sides to whatever was for that politician the best at that moment.