Many thanks again for your input! But I must reiterate: I have an obj export of this mesh that looks perfectly fine in-game, the lighting of which matches what I can see in blender. So I don't think there's anything I can change on the geometry that would fix my issue.
Nevertheless, I've tried tweaking all the parameters I could find in the export settings, and also removed the caps at the top of the middle pieces, but nothing seems to do the trick.
I think it's a reasonable assumption to think some piece of the import/export pipeline is to blame, but I don't really know what else to try.
I've tried changed illumination settings around, and I haven't found a setup that fixes the problem. Take this shot as a comparison (first is the dae export, second the obj export), you can see the directional light as well.

There's also this one I took where I replaced the environment with a very basic one, with clear color (no sky), zero sky contribution (just in case), and white ambient light with a directional light, just to rule out the possibility that it's a sky reflection:
