I got around the problem by changing my parenting structure.
What I was doing before looked like this:
Model
|
Actor
|
---------------------
| |
Exposed Joint Exposed Joint
| |
Model Model
Now I’m creating my model tree like this:
Actor
|
----------------------------
| | |
Model Exposed Joint Exposed Joint
| |
Model Model
So that none of the things with textures on them are ancestors of other things with textures. It works, but I still think there is/ought-to-be some way to prevent texture inheriting.