They both work. Personally, I would use "were".
We broke up because I didn't think we are/were a good fit.
Are both useable? What's the difference and what would a native speaker use?
You're not breaking up now. Since the break-up is in the past, the reason must logically be in the past, too. You didn't break up then because you're a bad fit now. You broke up then because you were a bad fit then.
If you're trying to make the point that nothing has changed and you're still a bad fit, then you might say, "because I don't think we're a good fit."