How can a church exist across the multiverse with different perspectives on its god without causing a schism? What's the best way for it to maintain control?
As described, it's unlikely.
In realm C, [G-]d is a single monotheistic deity.
This part right here is the problem. You can have a vast amount of religious pluralism and syncretism among polytheists. Some will draw all the clergy under a single wide tent, viewing the myriad gods as facets of the transcendent in the manner of India's Dharmic faiths; others will maintain separate clergies, traditions, liturgies but consider their various storm, fertility, virility, love, spice, toilet, and wombat gods and goddesses interchangeable in the manner of the Roman 'Greek interpretation'.
In either case, syncretism will cause them to share various traits but you'll still get the variety of practice and doctrine that you were looking for.
Exclusivist monotheism and hostile ethnic faiths are precisely who your clergy would be defending against, at least from the inside. The heretics to the greatly tolerant are the intolerant who insist upon their own way and refuse to respect and play ball with the rest. Witness the Roman treatment of the Jews (ethnic monotheist partisans of 'Jupiter') or Hindu spats with the Buddhists (whose founder rejected the importance of gods to personal salvation and became a Hindu avatar of deception and Satanic faith-testing for his pains) and Muslims, who permit predecessors of their faith for a fee but have religious injunctions against permitting idolatry.
Thing is, though, intolerant faiths are much more marketable to powerful secular rulers. If they have enough adherents to ride out the civic unrest caused by changing faith, their tenets give such rulers complete permission to exploit their power against weaker neighbors in the name of spreading the One Truth and immortality as a holy men for their troubles. People around the world still praise and remember David, Asoka, Constantine, Charlemagne, Muhammad, Vladimir, & al.
When such faiths are strong, even such notionally pacifist strands as Christianity and (e.g. Japanese & Sri Lankan) Buddhism will feel a divine right to expand; indeed, the clergy will tend to speak of a divine duty to expand, to save the souls of the as-yet-unborn children of the heathens and benighted. When such faiths are weak, like the Jews under Rome or the Palestinians under the Jews, they will view their neighbors as illegitimate and resistance as a test of their faith.
Plausible options include
Such faiths existing and in uneasy pluralist participation with the rest because we're coming in just after they've burnt themselves out on decades to millennia of crusades, jihad, and wars of religion. You'll see 'culture wars' similar to modern America as stalwarts aim to resist tolerance of everyone else's 'decadence' and 'immorality'
Such faiths exist as nuisances, terrorists, and wasteland enclaves (think Utah) but are not part of the interplanetary religious order you're describing. The official faiths might suppress them as a matter of policy or attempt to ignore them in the name of upholding personal conscience... until a certain threshold of insubordination or political power is reached, at which point the gauntlet comes down (think Roman Judea or the entire religious history of China).
Revealed, exclusivist monotheism as such does not exist or has just begun, but some philosophical monotheisms on the Socratic or Deist model exist, either as the usual mindset of secular university faculty or as the leaders of separate realms. They would consider their single deity to be identical with existence (or nearly so) and accessible to trained reason; they would consider their polytheist neighbors less refined; but they wouldn't see any need or sense in going to war over the topic.
The polytheists are just right. (Hey, those gates didn't build themselves and you're talking about their being maintained by clergy, not engineers.) In your universe, there actually are divinities or divine emanations and it's unquestionable that they have and provide spiritual power (basically, magic) and the monotheists, if they exist, don't or are just on par with some of the more powerful gods. They could even spring from separate vain and upstart gods (basically, Lucifer).
I'd consider that some major tensions in your societies are going to be how monotheists fit in (if they exist); how the clergy deal with the secular and military powers in their realms; and, if they are the secular and military powers in their realms, how they avoid using that power in ways self-destructive to the social order.