What about the Romans exalting their relationship to the
Sibylline Books? Even in OTL the books had immense power over the inhabitants of Rome. This went so far that they sacrificed humans because the Book (or rather the guys interpreting it) had suggested they should.
So my proposal:
1. Change the legend from Tarquinius Superbus (last King) buying them to, let's say, the Sibyl donating them to the Roman cause after the overthrow of the monarchy. Load it up with some myths about Anatolia and Trojan influence.
2. The books become a figurehead (in that regard akin to the republican
Rex Sacrorum) for the no longer existent kingship. However, the priests caring for them derive some political influence from the prophecies.
3. Over time the real political system changes to something like the OTL Roman Republic, with the books stil maintaining significance.
4. Now some Alt-Caesar comes around. He also traces his lineage to Trojan origins and who, after an Alt-Civil war derives a weird form of government making him a custodian of the books, which are instituted as the formal ruler. Offically the books through the custodian hold executive power, while republican institutions hold legislative power.
5. Over time the system evolves into an Alt-Roman Empire with the Sybeline Books as its offical head.
This results in a ruling inanimate object that is even more than a simple figurehead, because the prophecies were actually followed through. Abrahamitic religions will get hard though, if their orthodoxy doesn't get a lot more open towards syncreticism than iOTL.