Cura messes up my print. How can I make it print what I see in Blender?
It's common for modeling software to flip normals, causing what you describe. You said,

I only care about the externally facing faces.

But are you sure they are truly externally facing? Turn on normals in Blender and ensure that all the arrows/pointers are external.
Another problem that can generate the results you're seeing are self-intersecting surfaces, non-manifold edges, and a couple others that are explained in a 3dhubs article. Correcting those problems involve a bit more examination and trouble shooting.
The slicing software prefers to have clean, well defined triangular surfaces in order to create a print file. When something is amiss, you get results you've presented.
With the STL in hand, I opened it in Meshmixer (free) and immediately was given a clue.

The zebra-stripes are reversed normals.
Here's another error point, difficult to see clearly in flat 2D image, but panning and zooming and orbiting and all those 3D manipulating things shows there's nothing there:

The other end of this not-joined area shows a bit better:

After selecting all I could find/locate, I did the reverse normals and here's a slightly improved result:

Unfortunately, the other borked up portions prevented meshmixer from performing a suitable auto-repair, completely destroying the model in the process.
Those joint corners and any other colored lines have to be addressed individually.