I absolutely adore the name Elise, but IMO they’re really just two separate names.
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Maybe others have had more luck than me at ‘engineering’ nicknames, and if you CAN engineer nicknames then I don’t think it’s too stretchy at all.
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But for me choosing the nickname didn’t really work - once you’re dealing with the day to day busy-ness of parenting, changing poopy nappies and rushing between activities etc, all sorts of random things pop out your mouth 
and then there’s whatever other people choose to call your child too.
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I had planned my daughter Robyn to have the nn Bobbie, I’d heard it as a nn for Roberta and thought it was pretty cute. But in reality she gets called Bobbin, Bobster, Robbie, Robs, Robster-Lobster, Robbie-Roodle-Chicken-Noodle, etc… unfortunately although it’s not my fave, Robbie has become the most commonly used nn (that I’ve heard friends call her) because starting with the R like Robyn is just more intuitive.
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So I don’t think Elise will stick unless you’re constantly consciously making the effort to call her Elise instead of Elizabeth and then it’s kinda like… well, why name her Elizabeth at all if Elise is what you’re wanting to call her?