Honestly, I think one of the best ways is probably the federal republican model under the Emperor. But to get this to work you need a way to essentially vet the Senate.
I introduce to you... A Roman Internal Affairs. Quite literally an anti-corruption campaign by name, senatorial blackmailer by action, that acts independently (in theory) of the Emperor, and put forward corrupt senators to magistrates.
After the Senate can be 'Trusted', then it can be used to rubber-stamp inheritance, until it is seen as legitimate. Perhaps even debating and rejecting a candidate or two.
That at least resolves the inheritance and civil wars - someone must be approved by the senate. Spin that out into regional candidate councils that can become local senates that put forward candidates for inheritance.
The problem is that you need at least an Emperor who recognise the value of a stable succession, and the need to institutionalise it. Essentially realising that Julius Casear and the two Triumverates were a mistake. These need to be avoided with proper institutions.
Perhaps the successor to Aurelius is both warrior and administrator, and recognises that whilst he's very powerful, if someone 'wrong' had the throne, it would be very bad, and since it doesn't impact on his direct rule, sets up these institutions, and more as he gets older.
1) It helps him to undermine the senators and other political figures that are corrupt, giving him more control
2) Since the only power he is giving up is succession, and has corruption charges as blackmail to use against Senators, he can use them as a rubber stamp at this point.
3) Recognising that the obvious flaw in the proposal is the army - as such whilst the Senate decides, it must get official support from either the Emperor, and/or the Army). This way whilst the Senate vets the candidates for their shortlist/preferred option, if neither the Emperor or the Army will stand for them, they have to go back. This should prevent usurpation in most cases. In case of no designated heir, the Empire would be led by two consuls as of old - one in the field, and one in Rome, alternating as suitable.
After this is set up, the control over the succession is a slow-moving but powerful political tool. Any other institutions he sets up (i.e. the selection councils, army approval system) can be under the premise of "No more Triumverates"
It is a bit of a step back, and besides the anti-corruption arm is almost stepping back to having a Senate with a Permanent Dictator in play, but in my naivety about the powers of the Senate at this time, this seems to be the best approach. Essentially, an Emperor who fears his own awesome power in the hands of another.