Ok, finally I got to the bottom of it all!
Basically, Blender 2.8x is all good to go with FBX LODs as-is, without any modifications. However, I made a modification to FBX exporter I totally forgot about - to make it not exporting Armature as root bone. The way it was done (either someone suggested the fix or I found it on the Internets :rolleyes: ) was interfering with export of Empties. Which resulted in screwed up FBX file when it comes to LOD groups.
All good now (that’s what I said last time, didn’t I :o ).