Allow.
From your post, it doesn't sound to me that you have been using this deity Taiia in your current campaign setting, right? So this will be the first time she appears in your game... then why do you even care if the PC is "canon" or not?
I would understand if you had used it already as written in that book, and you wanted to keep consistency. Then there may be good reasons to disallow (but not necessarily) the exceptional character.
But if you haven't used it, and you are willing to add a whole new religion to your fantasy world, it doesn't have to be exactly as written. Anyway, even in that book, Taiia is just an example. You could simply come up with your own Taiia with whatever variations you wish.
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As for scribing scrolls, IIRC those domain spells definitely count as "divine" even if they are "stolen" from the Wizard list. I seem to remember that there were in fact discussion ~10 years ago that those scrolls would normally be usable only by clerics with the same domain (or another domain granting the same spells), exactly because other clerics don't have them in their list while wizards can only use arcane scrolls. That's the RAW, but you're free to handle it differently.
Our friend was saying that his character could Scribe *ANY* arcane spell he could get a source on.
He said that he could scribe from another scroll, for example. I pointed out that he'd end up burning the source scroll in creating the new one, if it was to be allowed at all. The rules say it has to be a spell the caster has prepared, and that scribing it uses that prepared slot.
Next he'll try to argue that he could prepare Scrolls from a spell book, even if he can't actually prepare the spell himself.
I can see where he's trying to get to, and I don't expect him to stop trying any time soon. I just have to be the bad guy, keep an eye out for him trying to slip one through, and keep saying "No".
As for your "Allow" vote: Just to make sure I understand: He wants to bring a deity in that doesn't fit the game world (we're pseudo-historical, so Greek, Roman, Egyptian, Celtic, Norse etc all fit. Deities out of non-historical settings don't.)
Further the deity he wants has two separate Domain lists, one for each aspect, and he wants to pick one from each list.
The fact that the deity has no foundation or history in our world is a reason to disallow, as I see it. Not sure how you got to your view that that's a reason to allow it.
And allowing him to play fast and loose with the rules for the use of this deity, just because the "by the book" rules haven't been used before? To me, that's a good reason to enforce the "by the book' rules.
But then, I'm pretty much a "rules guy". A written rule may or may not be right in every situation, but it tends to be consistent and inarguable. It's a standard every player has equal access to, that they (and the DM) can count on being in place when making plans.
(There are few things that tick me off more in a game than when I come up with a tactic, based on the rules, and the DM decides to let me commit to it, then shoots me down because he's decided he doesn't like that rule.)