You raise an interesting issue, whisky. The Wikipedia article on
English terms with diacritical marks says (footnotes omitted):
In words of German origin, the letters with umlauts ä, ö, ü may be written ae, oe, ue. This could be seen in many newspapers during World War II, which printed Fuehrer for Führer. However, today umlauts are usually either left out, with no e following the previous letter, or in sources with a higher Manual of Style (such as the New York Times or The Economist) included as German.
The end of this passage is a bit unclear, but I think it means that the usual practice today is to simply drop the umlaut, unless the letter is printed with its umlaut.
Führer can be written in English with a lower-case
f, as a common noun for a dictatorial leader, and so the word is accepted in Chi, with both possible no-umlaut spellings,
fuhrer and
fuehrer. I'm finding it hard to think of other words in the Chi lexicon that had an umlaut in their original German.
Doppelganger/
doppelgaenger are too long.
Flugelhorn is not too long for a ten-letter puzzle, but the possible alternative
fluegelhorn is too long.
Uber was added as a word a few years ago, but I know for a fact that
ueber was not included, because it never occurred to me or anyone else.
As far as
roesti specifically is concerned, it is given as an alternate form to
rösti by the Shorter Oxford and in the Dictionary.com entry based on the Random House dictionary. And it is used still in publications. The Canadian CBC News website in December 2016 had a recipe for "Salmon tartare and potato roesti". My local newspaper, the Melbourne
Age, uses both
rosti and
roesti, with the former being about three times as frequent. (Presumably the word is not covered in their style guide, if they have one.) No umlaut appears that I can see, at least in the online version.
So I think the evidence is there for adding
roesti to our words.