If 운동하러 isn't correct, why is it one of the first options?
is 을 really necessary here?
Why is it haebwa?
It doesn't exactly have to be "운동을 해보다/해봐요." But 해봐 basically means "to try," or "to try and do", and when paired with "운동", it means "to try exercise" but if you combine the two words, you get "운동해봐요," which basically means the same thing.
It could also be -운동하부요?
You can use 운동하다 to form the compound verb, but you need to correctly conjugate:
Yes, I answered with 운동해봐요, but it was considered wrong?
Report it if it's still marked wrong.