Well, for everyone concerned at this point, Rome was a republic.
You may find the parallel between IVth and Vth french Republics somehow interesting for what matter political take on the clearly important changes : it was still the good ol' SPQR, only with a strong man that will prevent all these squabbling factions to mess thing up and to take on the average citizen.
The monarchical idea, with a vertuous man ruling above fairly and equally, was popular as much on senatorial/equestrial elites (especially with the growth of stoicism) than the people, and partially explains why the idea became popular and continued to be so even after Augustus.
So, assuming Augustus does this, you'd have not only dynastical concerns (which means military concerns, giving the army was one of the pillars and main supporter of dynastic succession) but also political and social.
Let's say the Senate enacts this, you'd have at "best" a return to the joyful situation of Ist century BCE : tensions, riots, maybe Civil Wars; before something decant from this.
At worst, another strong man appears and takes the upper hand on SPQR thanks to its institutional role and relation to the deceased princeps.
It would certainly change things, tough, would it be only when it comes to the percieved political and institutional role of the emperor, and maybe a more formalized and dualic relationship with the Senate.