I'm not Jeremy but I can answer this. Both instances of jIH are necessary. What you have here is two separate sentences, tlhIngan jIH I am a Klingon and SuvwI' jIH I am a warrior, joined by the sentence conjunction 'ej. Whatever is on each side of the 'ej must be a complete sentence. Just tlhIngan all by itself is not a complete sentence, so it can't be by itself on one side of the 'ej.
What you COULD do, though, is use the noun conjunction je. I don't know if you've encountered it yet. It is used to join nouns in an and relationship. It goes at the end of the list of nouns. So you could say tlhIngan SuvwI' je jIH I am a Klingon and a warrior. The phrase tlhIngan SuvwI' je means Klingon and warrior.