I guess it makes more sense if you think of the past tense response to the question 'How did you go on the exam?': i.e 'I went well/badly'. That would be the most common way to phrase the question/response - in Australia at least! And yes we are native English speakers too... ;) Maybe a Brit can back me up here...!
“He went badly” sounds wrong to me as a Brit too, but if it’s valid in Australian English, it should be an acceptable answer.
Duolingo’s English shouldn’t be only American English, it should just be English, and in fact there are already many answers that are not American English. Just as there are (as there should be) answers that are European Portuguese rather than Brazilian Portuguese.
I tried "He has done badly", which was marked wrong. No objection, as "He did badly" would seem to be a better translation. However, in response to Australian queries, in British English one would never use "He went badly" or "How did you go" in this context. Rather "How did you do" or "How did you get on" in the exam