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I don't think it needs a nickname. Eloise is a gorgeous name and if that is what you want to call your daughter, call her just that. If something like Ellie develops naturally that's cute too, but don't force it if you don't have to. As for a middle name how about:
Eloise Brida
Eloise Hope
Eloise Freya
Eloise Mira
Eloise Sabine
Eloise Matilda
Eloise Georgiana
Eloise Yvonne
Eloise Victoria
Eloise Felicity
Eloise Norah