There are probably a few ways to handle this, but I’m guessing the fact that you’re using “/partners” in your taxonomy rewrites is messing things up for using “/partners” alone for the CPT.
I think I would probably use “/partners” for the CPT only, and then define a custom rewrite rule that determines what the parameters after the various slashes correspond to.
If you take a look at the examples on the reference page, you might get an idea of what I mean. Specifically, look at the one from Akira Tachibana on that page.