I'm trying to find a German word that is the same as »Schadenfreude« but also has a tinge of empathy in it. Is there such a word?
There is the so called Schmunzelfehler which is an euphemism for a shortcoming in some good which doesn't really matter. For example a scratch on the backside of a cabinet.
It comes from schmunzeln, which means "to smile over something bad", and Fehler (error, mistake, failure).
Ihr Gesang war kaum auszuhalten.
— Oh, im Vergleich zur Band war das doch nur ein Schmunzelfehler.
I couldn't stand her singing.
— Oh? Compared to the band it was merely a Schmunzelfehler.
I assume you are looking for a word that describes reacting to someone's misfortune in a humorous way, without the implied negativity.
I would suggest Stichelei. It is the German word for a taunt. (Note that this word can not be used for non-verbal reactions, a laughter for instance. But then again, the same is true for the English word.)