Consider creating a publish script for both the debug_extension and dwds

We have forgotten to update the `client.js` and the `background.js` before publishing in the past. Ideally, these files are not checked into the repository but are built as a part of a deploy script.

Locate the publishing or deploy entry point for debug_extension and dwds, along with the generated client.js and background.js files. Check how these files are currently updated before publishing. Done means a repeatable publish script builds both artifacts during deployment so they no longer need to be checked into the repository.

Written by the indexing model from the issue text.