Circlet Of Persuasion does not seem to interact with Noble Scion (War) (switches initiative to charisma), Sidestep Secret (switches reflex to charisma), and Lore Keeper (switches Knowledge to charisma)
From the SRD regarding checks:
A check is a d20 roll which may or may not be modified by another value. The most common types are attack rolls, skill checks, ability checks and saving throws.
From the SRD:
An initiative check is a Dexterity check
This delicately engraved silver headband grants its wearer a +3 competence bonus on Charisma-based checks.
Lore Keeper (Ex): Instead of encyclopedic knowledge, you learn most of your information through tales, songs, and poems. You may use your Charisma modifier instead of your Intelligence modifier on all Knowledge checks.
Sidestep Secret (Su): Your innate understanding of the universe has granted you preternatural reflexes and the uncanny ability to step out of danger at the very last second. Add your Charisma modifier (instead of your Dexterity modifier) to your Armor Class and all Reflex saving throws. Your armor’s maximum Dexterity bonus applies to your Charisma instead of your Dexterity
Scion of War: You use your Charisma modifier to adjust Initiative checks instead of your Dexterity modifier.
basically a circlet of persuasion is a "poor man" version of a headband of alluring charisma. It will not affect non-d20 rolls and passive stuff but any d20 roll that uses charisma as its basic modifier (and is not "enhanced by charisma", so something like a paladin's saving throws will be unaffected) will benefit immensely by it.
In the case specified above an oracle of lore character (the oracle of nature version only boosts AC and CMD which are not d20 rolls) with the noble scion bred for war feat should see that +3 on knowledge checks, initiative and reflex saves. (as well as spell level checks and whatever else)
Edit: “Circlet Of Persuasion” is detailed in cr_equip_magic_items.lst line 101, the code just lists the basica abilities without taking into account if they are based on charisma or not after the original base modifier has been swapped. This might also be due to the fact that “swapping” a base modifier is currently handled as a bonus equal to the new modifier to which the old modifier is subtracted.
Microsoft Windows [Version 10.0.18363.836]
Openjdk12u Jre X64 Windows Openj9 Windowsxl 12.0.2 10 Openj9 0.15.1