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Apparently there is no way to build in CSS with SSR support. I also ended up realising that this is not ideal. Most big libraries use separate CSS.
My final solution is just setting up vue-cli-service to build the code as a library like so.
vue-cli-service build --target lib --name myLibName *.js
And no need to change anything in config file. It is all done out of the box, unless you want something special.
Cheers.
Source:stackexchange.com