Elsie would be a completely wrong way of transcribing Else in most Germanic language.
Elsa would be closer.
Or “Elseh”.
Elseh I'll grant, but it feels a lot less natively English, and you can't count on an American clerk or census taker to care about accurate transcription. It may have been closer to "that's not a name in English. Pick either Elsie or Elsa".
And yeah, Elsa was common too.
Elsie sounds more like a German diminutive or pet name of Else than a strict transcription.