I've spoken with one of the developers of Valkyrien Skies 2, in turns out that its physics thread indeed must be notified of when the server's paused. They linked me to the snippet of code of where this is handled in single player:

https://github.com/ValkyrienSkies/Valkyrien-Skies-2/blob/cf2626161b3185c4e337ac3904778859727894dd/common/src/main/java/org/valkyrienskies/mod/mixin/client/MixinMinecraft.java#L122

With the following suggestion

suggest to use VSGameUtilsKt.getShipObjectWorld(server) as opposed to the IShipObjectWorldServerProvider that this method uses

Try 1.3.2. I've tried to make sure VS knows the dedicated server is paused when I do pause the server and from my small tests it doesn't flood the console anymore so I assume it's working fine. Let me know if there's any issues.

Seems to work for me!