"František hurt Kateřina with that word."
Translation:František tím slovem Kateřinu zranil.
That's correct grammatically, although a little difficult to imagine in practice, because "tamto" is mostly used when you are pointing at the object (or you at least can point at it). If I wanted to imagine a scene where I'd use this sentence: There's a blackboard in the room with some words on it written by František and someone is asking me "What has František done to Kateřina?" - then I can point at the specific word and say "F. zranil K. tamtím slovem."
It isn't. "František zranil Kateřinu tím slovem" is one of a few hundred accepted answers. I don't see any recent report though so I can't tell you if you made a mistake/typo there somewhere.
Also be aware that your solution stresses "that word" - we already knew F. had hurt K., the new information is that he did it with that word.