If you have developers on staff you could talk to them about setting up a static publishing pipeline. Obsidian is just static files and can be placed into a git repo, and there’s even a plugin that automatically commits all changes to the git repo on a set time cadence for you as well, or you can commit manually.
The pipeline can be configured to run on every commits, only certain commits, etc. When the pipeline runs it would run the Markdown files through something like Jekyll to build static HTML pages and then publish them to whatever server is approved. It could even in principle be configured to publish only files that have particular tags, or are in a particular folder, etc.
This should be something relatively easy for a mid-range dev to put together, since they usually do more complex versions of this for their coding projects anyway.
Since everyone’s environment is different its not reasonable to come up with a single turnkey solution to this, but the building blocks are there for devs to put together pretty easily.