I spent some time looking over the demo project you linked and I'm not really sure how it's supposed to help me out here. The room that uses LODs is being loaded from a file. Any information on those LODs is buried in there. I'm using an ArrayMesh I'm compiling myself. Literally the only code exposed here is pressing the L key to toggle the Viewport's MeshLodThreshold property.
To clarify, the other levers I have identified that I have to pull are the Viewport's "MeshLodThreshold" property and the MeshInstance3D's "lodBias" property.
The LodBias property is supposed to force the lowest level of detail at 0, so I'm setting LodBias to 0 on my MeshInstances as an experiment (and it's still using the higher detail indexes). Similarly, a high MeshLodThreshold value is supposed to force a lower level of detail, so I'm setting that to 1000, which also isn't having any impact.
In addition to visual inspection, RenderingInfo is reporting the exact same number of primatives when I toggle LOD on and off on the Viewport.
I have gone through and manually applied my lower detail index arrays as the primary index array for the ArrayMesh (not using LODs) and have confirmed that they are set up as intended.
I feel like there must be some top level property or Camera property that I'm missing here but I don't see anything in the documentation.