By experimenting, I found that significantly lowering the (apparently undocumented) dataref sim/private/controls/lights/legacy_luminance_spills to values around 5–10 helps a lot. It does increase the already unrealistic light bleed from the overhead lights into the cockpit, but IXEG behaves like this anyway, and those lights can simply be turned off (at least in flight).
You can set it, for example, via a FlyWithLua script:
set("sim/private/controls/lights/legacy_luminance_spills", 5)