I love this nickname but not a fan of [name_f]Elisabeth[/name_f] or [name_f]Elspeth[/name_f].
[name_f]Do[/name_f] you think [name_f]Elsie[/name_f] could be a nickname for -
[name_f]Felicity[/name_f]?
[name_f]Anneliese[/name_f]?
Emeliese?
[name_f]Alessia[/name_f]?
[name_f]Giselle[/name_f]?
Name combinations of LC initials like [name_f]Lorelai[/name_f] [name_f]Calista[/name_f]?
Any other names to get the nickname [name_f]Elsie[/name_f]?
annaliese works imo! 
The only one that I think doesn’t really work on your list is [name_f]Alessia[/name_f].
[name_f]Estelle[/name_f] and [name_f]Eloise[/name_f] are my favorite options.
[name_f]Lucille[/name_f] [name_f]Cornelia[/name_f]
[name_f]Laura[/name_f] [name_f]Caroline[/name_f]
[name_f]Lydia[/name_f] [name_f]Cosette[/name_f]
[name_f]Lucy[/name_f] [name_f]Cassandra[/name_f]
[name_f]Lavinia[/name_f] [name_f]Carys[/name_f]
[name_f]Giselle[/name_f] works imo
[name_f]Anneliese[/name_f] too, and [name_f]Alessia[/name_f].
I’d also suggest: [name_f]Elswyth[/name_f], [name_f]Estelle[/name_f], [name_f]Estella[/name_f], [name_f]Eloise[/name_f]
Might be a bit of a stretch, but I love [name_f]Elsie[/name_f] as a nn for [name_f]Demelza[/name_f].
I could see LC initials working. The others are far too stretchy. How about:
[name_f]Elsa[/name_f]
[name_f]Eloise[/name_f]
[name_f]Elise[/name_f]
[name_f]Eliza[/name_f]
[name_f]Elodie[/name_f]
[name_f]Elsinore[/name_f]
I love [name_f]Anneliese[/name_f]! 