Conversion by itself would mean gaining enough popular support in the target system for it to surrender itself to you. By its nature, this is treason against the host country, which in turn means that this process would have to be covert, and thus unknown to the owning faction, much like a Hack. There would still be the %bar that needs to be filled, that part we still need - however, for that to be put to use, you would need agents on the ground - AKA Sleepers. 

Like with current Sleeper tiers, you would need a certain amount of Sleepers on the ground to perform Espionage actions, which would practically function as a Hack that starts and ends in the same location, you just use different variables to resolve it, like Happiness (high hinders espionage, low supports it), garrison size, military facilities in place, Hero presence, and so on. In accordance, to convert a system, you would need several Sleepers. To maintain hold of a contested system, you need to keep them very happy and heavily garrisonned, and if you're really worried, plonk a Hero there as well. That combo should require the direct intervention of RNJesus to crack.

Peaceful Conversion is exactly what it says on the lid: if the government type is Republic, Democracy, or Federation, then they'll hold a vote (which both sides could influence with either Dust or Influence), which will decide if they'll join you or not. Should they decide to stay, a set cooldown will be put on the vote to prevent spamming the vote and turning the whole thing into an auction. The AI should pursue this version if it wants to avoid a war with you. Obviously, this isn't an option against a Dictatorship or an Autocracy.

Forceful Conversion means that the decision is not made by a vote but by a land battle, where both sides are on the ground. Each group's size is determined by its support, drawn from the system's garrison. Naturally, in exchange for some Dust for each point of Manpower, both sides could deploy additional troops from their reserves (or without paying for their trip if it would cause a balance issue). Neither side could use the Draft action during the land battle. Whether or not the rebellion fails, the system simply loses the appropriate number of troops in its garrison, and suffers a turn of Anarchy, during which it produces absolutely nothing.

Uniquely, the Nakalim could "buy" Sleepers in systems affected by a Temple with Influence, who would have limited available actions compared to the others but they would be somewhat harder to catch (representing how a missionary works differently than a true spy). This way, the Nakalim would not only not have to rely on Hacking to convert, but would also be relevant in the espionage business (the Vatican comes to mind), again without Hacking, which is the Umbral Choir's shtick (who can't convert systems already, so there's no way for them to abuse their Sanctuaries, even if it would be tons of fun). Additionally, Hacking itself would be more desirable for non-UC factions - however, in doing so it would be nice if the cybersecurity aspect of the Hacking mechanic was more clearly communicated and straightforward (like knowing that a higher level System is more likely to discover your Hack and start a Trace, stuff like that).

I understand that patching is more complicated than just throwing ideas around. I also understand that a solution must be principally correct before you could work out its exact execution. In that spirit, I believe the above solution is principally sound, while also working with mostly existing assets, and even what needs to be added principally already exists. 

Then again, I'm not a video game dev, so I could easily be wrong.