"Sorry, thank you!"
it's too colloquial. You can say it to your friend and it doesn't express a high degree of regret.
From watching a native Hungarian speaker on youtube teach a few lessons, she said you could just simple say ''kösz'', is this not totally allowed or?
They are colloquial terms to express "thanks". We normally don't use them in a formal conversation. E.g. a student should never say that to a teacher otherwise he or she would show total disrespect.
When I gave: Elnezest kerek, koszonom! my answer was rejected. Why? is it over the top to say kerek as well or does it change the meaning?
The other proposed answers are unnatural and illogical, that they hardly pose any challenge. ("Sajnálom, ide szépen!" and "Menj, köszi!")