Yes, "becoming tired" sounds unnatural. The situations in which "become" works are much more restricted, so better to use "get".
Also, when kids talk about what they would like to be later when they grow up, you would use the word "be", not "become".
Or, for example, what are you going to "be" for Halloween?
I think the situations in which you would use become are maybe a little more abstract, EG it's becoming clear that she has ulterior motives, or it has become impossible to operate in the sphere of politics in these days of fake news.
Not totally I sure I agree that kids wouldn't say "I'd like to become a racing driver" or "I'd like to become a fireman". The formality of 'become' generally does indicate a kind of finality, though. A doctor describing someone on their last legs might say "She's becoming tired", for example I suppose.
Mentally I've been substituting "er ved" with "is about to", and at blive" with "is becoming". Thus the total translation becomes surprisingly specific: "the horse is about to become tired" - and this was accepted
My question: was my translation actually correct, or was duo being uncharactoristically generous?
Your answer is correct. :)
At blive is "to become" or "to be/stay". Ved means "by" or "close to", but in combination with a verb it's more like "about to" or "happening right now". All in all, the interpretation of sentences like this is a bit more on the flexible side. "About to become" doesn't make much sense in most situations, but it's still a good translation.
It's a bit a weird construction, usually you say something like "The horse is becoming/getting tired." As David says above, "about to" wants to be used with events that happen in an instant, but getting tired is usually a (longer) process.
But then it is a bit nitpicky to accept "about to become tired" and not accept "about to be tired". So your translation is understandable, at least.