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In short, after struggling for an evening, I finally managed to find a solution:
1) Go to the material properties.
2) In the AlphaMode line, set it to AM_OVERLAY.
3) In the Priority line, set it to 224.
and voila, we are no longer bothered by overexposure.
P.S. But I encountered a small glitch in only one place - on the background of the small smoke of the dwarf city at a certain camera angle. Basically, this is already a minor issue. But, ideally, this method needs to be tested.
in principle, you can create an effect with such an owl for the map, but use the old effect in battle, or make a special owl without legs for the map.