It is not necessarily "complicated", you can take as a basis the religion system of CK3 with some adaptations.
Each “cultural / ideological / religious” doctrine costs a certain number of cultural points: military, economic, religious, diplomatic, societal and etc.
These points are generated by the pops of this culture (perhaps according to the ethics of the empire and / or of the pop), "cultural workers" and others, for example, with the civic "citizen service", the soldiers generate military cultural points.
With technologies, traditions, ascension perk and more, we can unlock new cultural doctrines.
Each empire according to its diplomatic and other relations generates a cultural pressure on the other empires. This pressure depends on the basis of the production of cultural point of the empire.
Obviously, he may have less passive means of spreading his culture, such as espionage or war for example; where the branches of megacorporation. Another peaceful approach would be, for example, to finance cultural promotion campaigns in another empire, but it requires authorization from that empire, the empire referred to could have certain benefits.
Ultimately, we could make our culture the galactic culture by a vote of the Galactic Community, this does not automatically adopt its culture to other empires, but greatly increases its cultural pressure. After that, we could possibly pass another vote to make other culture illegal ...
An empire can adopt another culture or reform a foreign culture that it would have adopted, obviously, it is not free, not without conditions and not without consequences.
Cultures could have a system of levels or parallel levels, one can progress in various branches, each branch being able to have different conditions of progress.
Obviously, we have to think about the system so that it does not duplicate the system of traditions. Each system would have its own mechanisms.
Progressing in a culture could also allow to unlock new civics (or more powerful versions for some already accessible, but possibly with penalties).