"Ela quis que ele tivesse desculpado."
You are probably right. I just have a little doubt though. Dictionaries I have looked at show that desculpar can be used non-reflexively too. Is it possible that this sentence could actually have meant something else (given you only saw the Portuguese) and is missing an object pronoun other than "se"?
Yes, it can also be used non-reflexively, but in that case it means "to take the blame away from" (pretty much a 'forgive') instead of apologize.
Tudo bem, eu te desculpo por sua falta de tato. = It's alright, I forgive you for your lack of tact. (notice that the most accurate word to use in this case would be 'perdoo', from the verb perdoar, 'to forgive'. It implies the person felt offended by the other's actions.)
Paulo desculpou-a por estar atrasada. = Paul forgave her for being late.
Notice that the above sentence implies that Paul knows the blame wasn't hers. He probably knows traffic was unusually intense that day.
When 'desculpar' means 'apologize' it is always used reflexively, as must be the case in this phrase, since Duolingo´s translation to it is "She wished he had apologized". This means the woman wished the man had made an apology (to her). The translation is: "Ela quis que ele tivesse se desculpado" (para ela).
In regards to your edit to make the question more clear, I think I get it now. It is definitely possible that Duolingo forgot to add the article 'a'. The sentence would be thus:
"Ela quis que ele a tivesse desculpado."
It would then mean "She wanted him to have forgiven her".
Desculpar can also mean "to excuse someone", such as in the sentence I just wrote to my mother after taking a long time to reply to her e-mail:
Oi mãe, desculpe a demora. --> Hi mom, excuse the lateness (in replying).
Even though the verb is in the Imperative form, it is not considered rude at all and is very commonly used. It is really just a faster and more casual way of saying "Desculpe-me pela demora" ---> excuse me for my lateness.