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It's not technically Versatile Performance that the Evangelist boons of the Deific Obedience feat grants, but rather an ability called Versatile Artist that works like Versatile Performance.
Versatile Artist (Ex): Drawing on Shelyn’s divine inspiration, you can substitute your bonus in a specific Perform skill for your bonus in another related skill, as the versatile performance class feature. Select one type of Perform skill which you can substitute for its associated skills. If you already have the versatile performance class feature when you obtain this boon, choose an additional Perform skill to substitute; you gain a +2 bonus on checks with this additional Perform skill.
That being said, I really don't see how you would read that text as Versatile Artist being anything besides a special version of Versatile Performance for devotees of Shelyn and thus meeting the requirement for the Persuasive Performance feat.
You'd have to be an absurdly literal stickler for the rules to not allow it, IMO.