There are actually two men who can't answer, not one, so the solution the way you proposed it won't work. Also, it assumes that not answering is permissable, so the random answerer can also choose not to.
"if B says yes, then you ask the L (the only one that lied) who the T is"
What if two of them lied?
I don't see how two men can't answer, as only one was asked an unanswerable question. And since the random answerer must either lie or tell the truth, i disagree that he could choose not to answer.