I think both constructions are grammatically correct, so you both are right, but when I read your use of “through” to avoid repeating “with” it sounded stylistically better. I mean it just sounded better to my ear, but the other sentence was okay too, probably so much so that few would notice.
I used to tell people on italki a sentence might be grammatical correct, but poor stylistically. That’s what English professors at universities really get into with native speakers. Not how to make grammatically correct sentences (maybe sometimes) but stylistically better ones. One thing I do at work, especially in revising things my clients have written, who are native speakers and send me things they write to file with the court, is re-writing their sentences for them to make them stylistically better.
One thing I learned early on is to avoid repetition. I heard someone say once (who was a PhD from Sanford) “well, I used THAT word/verb already once in the sentence before, so I can’t use it again.” I always thought that was a good rule of thumb, and I suppose that’s why languages have so many synonyms, for stylistic betterment. If you used the same words over and over again it would demonstrate a limited vocabulary and sound almost repetitious.
I have a client, an Air Force captain, with a college degree, and who is a native speaker, write up a declaration for me to file with the court. He did a good job of it, but wrote “because my wife hates me..” I told him it all sounded good, but to drop the “my wife hates me”. There’s something wrong with saying that, and it’s not going to sound good to the judge. So, I told him to say something different and softer, like `because my wife disapproves of me`, It’s stylistically better for the context. Does that sort of make sense? It could just be a legal thing too.
You could also say “I cope with diversity by determination.” Jerry’s suggestions is good too. But, it’s like any fitting preposition other than repeating “with” is stylistically better.