Much as I'd love a Feo Ul summon...Seraph as the "true" Lily actually makes a lot of sense. You also summon HER for SCH's L3 LB.
https://www.youtube.com/watch?v=AF2NrrJB8zA
(Hm...that video doesn't show it as well. Though the AST one there is amazing - it always is - especially with the music...)
Better video/timestamp to see Seraph: https://youtu.be/waTYGA99W4M?t=64
Can also see it in this one here: https://youtu.be/3PdR6PqEbyM?t=189
And the wide view here you can also see two smaller ones spiraling out, maybe representing Eos and Selene: https://youtu.be/3PdR6PqEbyM?t=204
Either way, I could totally buy that Seraph is Lily's true form.
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Though as to the "how would it work" - I'm not sure if Faeries are "constructs". How we summon them is, but SCH lore suggests that each SCH stone, when created, does so by making a PACT with a Faerie, which suggests that they exist independently and weren't created beings. Either that, or they were created but THEN were self-sustaining (enough) and had free will (enough to enter into pacts; and we see Lily have free will in some of the SCH quests, like preventing the killing blow on Curito) that the Scholars then made pacts with them.
Well...we made a pact with Feo Ul. It's how she's able to travel to the Source in a dream state to visit our Retainers and how she's able to hear us TECHNICALLY anywhere that we go and call on her. Which, by extension, should mean that she's connected to our aethertrail (how we're able to teleport back and forth between Source and First), and between that and the pact with us, we should be able to summon her whenever if she agreed to that. There might be a lore question about why she would only be able to take her smaller form and not her Titania form (easy enough to answer when not on the First, but when ON the First, she presumably should have access to the fullness of her powers), but between the SCH pact with Lily already and seeing how pacts with Voidsent and even Primals can work, I'm pretty sure there is more than a little argument for Feo Ul to be able to use Lily as a medium (provided Lily consented to all this) to come and help us out from time to time.