Working entirely off unreliable memory, Allag used the "modern alphabet". In fact, the "modern alphabet" is the Allagan one, or at least close to the Allagan one. (Source is EE2 p34-35.) This is a bit of a retcon, since EE1 p29 claimed the priests and scholars of the Fourth Umbral Era came up with an entirely new alphabet to hide their studies from the anti-intellectualism of that time, and that became the modern Eorzean alphabet. Now, EE2 says the Allagan alphabet is close enough to the modern Eorzean alphabet as to be mutually intelligible, except the pronunciations are different due to drift. How this affects historical research in-universe is not elaborated, other than mentioning how this language and alphabet was lost during the Fourth Eras and rediscovered later.
However, one handwave given by the lore writers (primarily Koji Fox) is that "early Allagan" and "late Allagan" are very different periods, since the empire spanned a few millennia. This is why the Labyrinth Of The Ancients looks nothing like Syrcus Tower, architecturally. This is also the explanation given for why there are many pieces of unfamiliar script scrawled on the various robots in Azys Lla: they're supposedly from a forgotten dialect of part of the Allagan Empire that is lost to history. From a Doylist perspective, the actual answer is this is what happens when the devs have to grab models and textures from FFXIII due to development crunch, and forgot to modify the Pulse writing.
So there's already a potential loophole for "this unfamiliar alphabet is Allagan, but not the only Allagan alphabet".
As for Solution Nine, the part which I'm expecting a lot of speculation from is how the alphabet used there is identical to the "proposed Ancient alphabet" combining Eorzean and Vrandtic alphabets to find a common ancestor. Putting aside the caveats about "this is what in-universe scholars think", the fact that it's used here gives a very high chance that this is the Ancient alphabet, with only a miniscule chance that it's all a coincidence.
So somehow, Solution Nine retains the Ancient alphabet, after the Sundering and twelve thousand years. This is remarkable, because even the First did not, despite the First having literal Ancient ruins that survived those twelve thousand years, as Anamnesis Anyder. So presumably there have been fewer world-shattering and culture-destroying events on the First, but Norvrandt still has a completely different alphabet. (I do not believe the alphabet will have changed in the hundred years since the Flood of Light; too little time.) So how did Solution Nine, on the Calamity-prone Source (including the "continent-breaking" Fourth Calamity) have writing in the Ancient alphabet?