alvordr's post in [Solved] 2021 - How to Export ? was marked as the answer
Yes, that works. This makes me think it's been so long that I'm literally forgetting how 3D Coat works, but so much has changed. I tried retopologizing 3 parts of my sculpt and unwrapping them. I hid everything else and then painted onto those three parts. When I click on Export Objects and Textures, I get the same issue I've had. What am I missing?