It's actually the shadowcaster, which is slightly different (it only happening outside should have been the clue to me) but still fixable! After a bit of tinkering, I think I've got the simplest steps down:
Only tools required are TSRW and Blender.
I used TSRW 2.0.86 and Blender 2.93, specifically, if anything in the screenshots looks different.
In TSRW, Create New Project > New Import, and select .package from the dropdown so you can open the CC directly.
In the Mesh tab, export the high detail mesh to .obj - I named it Reference. The project can be left open for now.
Open Blender, then press A then Delete to empty the viewport.
Go to Import > Wavefront .obj to import the reference- making sure that the import options are as so:
You'll know it's right when the mesh from buffetwood imports into Blender as two meshes called group_0 and group_1.
What you want to do is remove the pieces that aren't always visible. This one was easy to do from the UV layout.
In the UV Editing tab, make sure the viewport is set to Edit Mode and UV Sync (two arrows button, highlighted 3. in the screenshot) is on. All the extra decor is at the bottom, so you can just drag the selection box over all of it.
Then, with the cursor in the viewport press Delete and select Faces.

If some of the other tables have more complex UVs, you can also select the mesh parts by hovering the mouse cursor over them and pressing L.
Then you can go back to the Layout tab,
Object Mode.
Click on the mesh + shift-click on the dropshadow (and any other groups, if you edit an object that had more).
File > Export > Wavefront .obj, this time making sure the export settings are ticked/unticked in these areas:
Back in TSRW, import the .obj back over the main mesh. Any error at this point likely means the import/export options earlier were incorrect.
The mesh will lose its geostate, but don't worry- this file won't be used for the final product.
Switch to the
Shadow high level of detail from the dropdown list of meshes, press the blue refresh icon, and tick the box for the main group, then OK. This generates a shadow mesh using that one as a reference.
Export the shadow hlod to .obj - I named it ShadowHigh.
Then repeat these steps for the medium level of detail mesh + the matching shadow, if desired.
For the final stretch, you can close this project without saving. You'll want to import the original package again in a new one to restore the full meshes.
Switch to the shadow hlod and import ShadowHigh.obj - selecting
'No' to the popup about converting it to a shadow, because it already is one.
Lastly, switch to the main mesh and untick the 'ShadowCaster' flag from the Extra data.
This is the main culprit causing the invisible bowls to have shadows- and to be honest, I'm not sure why objects that have dedicated shadow meshes are assigned ShadowCaster to begin with. Just changing this isn't enough to fix this specific object because those shadow meshes had also been generated with the bowls.
Repeat for the medium lods.
To quickly export from TSRW, go to Edit > Project Contents, then Export > to .package.
Voila
Cardinal has been taken by a fey mood!