Hi Julien,
The best compromise between simplicity and control here is usually to create an entirely separate item listing in the Chrome Web Store. As you mentioned, you can use the "BETA" tag or similar to avoid a repetitive content policy violation. You can also use the various visibility settings (unlisted / trusted testers / published to domain) to control who can install the extension.
If you want more control, and to avoid the review process, it is possible to set up your own update server by setting up a web server with an XML file and the CRX files you want to distribute. This also requires policy on the machine of any users. It is non-trivial, so not usually what I would suggest, but I like to mention it for completeness. That process is mentioned in more detail here:
https://support.google.com/chrome/a/answer/9296680?hl=en