No idea of anything already made, but this doesnt sound "way too hard" to code one self if you know how to program.
Self-Hosted Main
A place to share alternatives to popular online services that can be self-hosted without giving up privacy or locking you into a service you don't control.
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We welcome posts that include suggestions for good self-hosted alternatives to popular online services, how they are better, or how they give back control of your data. Also include hints and tips for less technical readers.
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I'm not aware of a self-hosted option, but Slack is pretty great as an office IM tool.
Email could also fit this case if you configure it the right way
Matrix/Synapse.
I used a $0 tier Slack workspace for this in a small business, it worked OK. The Slack clients are enormous in memory.
IRC, it's ancient which may sound bad but it means that most implementations have grown mature and won't cause issues.
And plenty of web based clients for it too
You could probably use rocketchat. It does allow DMs, but there are also rooms you can add everyone to for increased visibility.
nextcloud is really great. you can store files or even video call them in addition to messaging. You can create roles message groups etc
We use signal and have a few groups for each category of staff. Super simple, and secure.