Would anything be wrong with "Is het een griep of een korts?"?
Yes, koorts is with oo.
Ah. Duolingo's complaint was that I added a second 'een'. Is there anything wrong with that?
Good question. Usually DL doesn't accept an extra "een" even though it doesn't change anything about the meaning. Maybe you could suggest it as a alternative translation and see if it is accepted.
Is there any way to suggest a translation now? I.e. while not in the middle of the quiz?
No, you can only do so during lessons.