| Mort the Cleverly Named |
I would say no. Sermonic Performance states it is "similar in all respects" to Bardic Performance, and even refers to the abilities granted as Bardic Performances. That, in this case, one of the abilities is a Bardic Performance granted by the Sermonic Performance class feature is of no consequence, as both are Bardic Performances and thus starting one cancels the other, even with Lingering Performance.
Yggdra
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The difference between them is that one is named "Sermonic" and one is named "Bardic". Since two Bardic Performances cannot simultaneously be in effect, I am on the side of the fence that they do stack with one another. Otherwise, why not have the Evangelist "gain the Bardic Performance feature" and limit the performances allowed from there? There is no reason to call it something different.
I see where Mort is coming from, though. "Sermonic Performance: Bardic Performance: Inspire +1" is still "Bardic Performance: Inspire +1" and- if considered that way- should not stack.
The problem is, you're multiclassing for this, and sacrificing a LOT to not have them overlap for a couple rounds. Two separate, non-stacking spellcasting classes and all of the fun sacrifices that comes with multiclassing, AND a feat spent, on top of losing spontaneous healing is plenty in my book to allow a player to swing the board in the party's favor twice as much. It wouldn't get very much out of hand, considering it takes them twice as long to get their Inspire AND Satire bonus increases.
Yggdra
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"A bard cannot have more than one bardic performance in effect at one time."
This implies that from one source, two bardic performances cannot both be in effect- regardless of whom the targets are. As this was published in the Core book, obviously it must not have had in mind the Court Bard
EDIT: let alone someone attempting to stack it with an Evangelist.