A few thoughts.
1. The State is not even nominally a democracy, and its equivalent of patricians and plebians is strictly informal (more like plutocrats and everybody else).
2. The State has no evident fear or hatred of kings, which was instrumental in getting Caesar killed. Tibus Heth basically already has imperial powers-- which appear to be on the wane along with his health and sanity.
3. You'd need someone with "authority" equivalent to Heth's to rise up in his place. There is no one like that on the present stage. Admittedly, Octavian didn't have that at the time, but we don't even have a Mark Antony.
4. The Caldari tend to pull together in a crisis, not apart, especially in the face of war with a foreign power and ESPECIALLY the Gallente. In order for a Caldari civil war to go hot, their borders would need to be secure.
5. That said, the Caldari have had at least one empire before-- the Raata. The resemblance to imperial Rome, however, is ... ambiguous. It may have had more in common with imperial China. If the megacorporations were to go to war with one another, become militarized and less profit-driven, and thus even more feudal than they already are, I could see the eventual victor (after, basically, a "warring states period") establishing something of that sort.
This is never going to happen while the Caldari have an active external foe to make common cause about. The furthest it is likely to go is deadly bickering among the upper echelons, possibly culminating in something like a second Morning of Reason.