Your mention of the Church's orders being integrated into the Empire as being ministries suggested to me that the Church is heavily integrated into the Empire. Much more integrated than in EOFS, where the Church is its own, autonomous entity that can crack down on proscribed techs. Of course, you may choose to have the Church have some autonomous military forces, but I'm more interested in the forces that the secular nobility are allowed to control.
Assuming this game is set in a pretty similar universe to
Fading Suns, the here's two possibilities that I could think of.
1) This game could take place in the aftermath of a EOFS game where the
Holy Writ was signed by whoever won the Emperor Wars. The Holy Writ essentially granted the Church major powers and influence within the Empire, and it's something the Church really wants...But the writ must have backfired by corrupting the Church and making it more involved and beholden to royal politics. There's not much point to have
de jure power and influence if you don't have the
de facto power and military that the nobility can provide. The Church has became so dependent on the Royal Houses for support that it started electing members of the royal families as their
Patriarchs (a fancy title meaning that you control the Church's votes and decide the Church's budget for spending on ministries).
You can still have "proscribed techs"/inquisitors* by having a puritan sect outside the control of the Church and the Empire monitor the Royal Houses closely. This sect could either be
Brother Battle (an elite fighting order) or
Temple Avesti (religious fanatics that are quasi-Inquisition). This sect does not care about whether the Royal Family they're fighting against is the Patriarch, and the only type of negotiation you can do is to give them maps. If you want them to stop attacking you, you stop violating divine law.
2) The game takes place in an alternate universe where the Second Republic was not formed by an alliance of merchants and The Church, but instead an alliance of The Church and the Royal Families. Here, the elected government is just recently formed, and it is trying to reconquer the whole universe after the First Republic was fractured (justifying the 4X). The Guild would be the previous rulers of the First Republic and was only tolerated because they have the ability to transport people through space.
The government system of electing Regents (or Doges?) is intended to be permanent, but it is possible to try to seize supreme power and abolish the voting system (declaring yourself Emperor) to centralize power into your hands.
I do like this idea too, but it might be harder to justify, and since I know very little about EOFS, it's possible I might deviate a bit too heavily from canon. There's also the fact that The Church's attitude towards technology was much more liberal during the Second Republic, so you don't have proscription. It's only during the collapse of the Second Republic that you see the Church banning technologies and supporting the establishment of the feudal order that we know of today. So it might not be an ideal solution if you want proscription to still be in the game, but if you don't want proscription, it's an idea to consider.
*On a related note, I'm not sure whether having a "NPC proscription agency" is even necessary if the agency needs to find the tech violation. Without a magical power to know where violations are taking place, the "NPC proscription agency" has just as much power as the players to stop things like Plague Bombs. It would make more sense then to have proscription would be handled by the players themselves, as they probably know the dangers of a player having too advanced tech.