nuxt-client-init-module | Provide client version of nuxtServerInit | Frontend Framework library

It's pretty hard to answer this without having the full project to build and run. I assume you're not using a separate webpack config file when building for production, otherwise you would have added that to the question. You seem to be using the correct syntax in the config file, so I'm guessing your css file paths are not quite right. They kinda look like the old way of doing things, you may want to confirm that they're not out of date. This old-ish github issue goes through the various things you could try, on of them being let nuxt find the "compiled" version of the css automatically for you like this: