Complicated that-clauses
You found a very interesting point. I thought about this for a few minutes, and I concluded that there may not be a straight-forward translation for this kind of sentences.

As you may have already known, in Chinese the adjective words/phrases/clauses are usually placed before the nouns they modify, therefore, if in your English sentence the relationship between your clause (A) and your antecedent (e.g. the man, B) cannot be translated into A 的 B, the thing becomes tricky.

If you want your translation to express exactly the same meaning, you can do things like:

我找到了他的包的（那个）男人 
我给他翻译了原文的（那个）商人

It doesn't sound very elegant but the meaning is still clear to Chinese speaker, I believe. We say this kind of sentences have a strong 翻译腔, because normally (at least long time ago), Chinese people don't construct sentences this way. It sounds like something translated from English.

If you want your translation sounds natural, you will have to rephrase your sentence, depending on the context.

我找到的包的主人 (The owner of the bag I found. Avoiding "whose".)

让我给他翻译原文的商人 (The businessman who let me translate the text for him. Avoiding "whom".) or 需要我给他翻译原文的商人 (The businessman who needed me to translate the text for him.)

I don't major in Chinese or linguistics, so I just answered from a native speaker perspective. Maybe there are some better way to keep both structural consistency and elegancy - I don't know that though.

In my opinion, if you learn some language, you not only need to speak it, but also need to think in it. If you always think in English first and try to translate your thought into Chinese, you fall into this kind of situations. That's exactly what happened to me, sometimes when I speak English.