Do keep in mind that Tetrimon have been branded heretics by the Theology Council centuries ago for opposing the Great Reforms and the imperial governement in favor of the old Council of Apostles.
As said above you won't likely change anything PF wise (and the players' inability to change anything in the lore or with their factions has always been a great complaint of all the RPers around). But the lore and the PF are an enormous source of raw matter for RP that can be exploited in an infinite ways possible. It gives us RPers incentives to create conflict or friendly relations (or whatever) with other RPers and factions.
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A good advice would also be to read the most important wiki articles on the eve wiki : the slavery article, and the scriptures article. They are about very fundamental bits of lore in New Eden and for an Amarrian character, it is a lot better to know about them or your character might end up making a fool of himself.
Yes, the chronicles are often hardly possible to take into account ICly since it is mostly about personnal stories and stuff that is obviously not known by everyone and certainly not our characters. But they are sometimes very informatives on the universe, especially the first chronicles (the ones on CONCORD, the Khanid, the Blood Raiders, Vak'Atioth, etc etc). These first chronicles are not about personnal stories but general facts, and fundamental ones. They can also often be retrieved on the eve wiki either, where they are copied-pasted most of the time. Then, of course, a lot of Amarrian RPers could advise you to read some very amarrian oriented chronicles like Chained to the Sky, that bring a lot of information on specific things of the Amarrian society.
On the Amarr subfactions, you have actually one of the most important number of all factions at your disposal. And if Amarr is not enough for you, you still have the greater Empire that is constituted of the Amarr Empire, the Khanid Kingdom, and the Ammatar Mandate, that have their own subfactions (but more player created even if they are only logical).
Inside the Amarr Empire you have 5 royal houses, and a sixth that has one foot inside and one foot outside (Khanid). Each of them, much like the caldari megas if I have to use a stretched analogy, has its own subjects, its own culture, its own mindset.
The Ardishapur are the most pious and faithful of all, putting religion above everything else and in the center of everything of their daily lives. They are pretty conservative on that aspect, but often represent what we like to call the Amarrian Orthodox, or the center/loyalists (and they do not like at all the new zombie Empress).
The Sarum family is a family of hardliners that cling to the Reclaiming of old. They are primarily an expansionnist family that still think that all of New Eden must be reclaimed with force like in the old days. Their part in the mighty Amarr navy is second to none.
The Kador family are one of the oldest and richest imperial houses (well, they were the richest before Tash-Murkon came and laughed at them). They are what we can call the most traditionnalist since what concerns them is not so much religion, but secular traditions and what has made the Amarr Empire glorious like in the old days. The social ladder, the pompous ceremonies, all of this is important to them.
The Khanid family fot exiled since Khanid II refused to kill himself after the succession trials when Heideran VII got crowned. Now they are half back in the Empire (they have a seat in the Privy Council of the families, but no voice). Khanid are very pragmatic and closer to the Caldari. They are probably one of the less religious houses, even if religion still plays a big role. But they merely use it as an excuse for slavery for example. Somehow, this house is like the Amarrian society without a lot of its morale and ethics, "tainted" by the practical/amoral (not immoral, amoral here) Caldari business mindset.
Then you have the liberal families like Kor-Azor, that has always been very close to Heideran and Doriam external policies, seeking peace and prosperity with other factions (and trying to enlight them more liberaly behind, of course...). Kor-Azor is the main model of the proud Amarrian mindset, but thinking like liberals. Strong and righteous, but with a liberal approach.
Tash-Murkon come from a minor family and their heir built her own fortune herself with a mining corporation. They are the richest family by far and own one of the richest corporation in New Eden. Their success and unorthodox methods bring them in the liberal side. They also are of Udorian origin, and not True Amarr. Their ties with other external factions are strong and their trade flourishing.
Then you also have all the national public side of the Amarr Empire if I can call it like that, which includes all of what belongs to the Emperor family, so, what belongs to the amarrian state. Like the Amarr navy, all the ministries, all the "public" institutions. They have their own flavor and do not belong to private houses.
And then, all the minor cults and heretics, like Tetrimon, etc. And also, the political factions of the Khanid and the Ammatar....
So yes, I think there is a lot of choice.
I am Amarrian
Good luck.