Multiple speakers used for my work ambiance in multiple rooms

Sorry, but there's a lot of unknowns in your post to best answer your question.

The short answer is: yes, having a multi-room speaker setup is possible and there are a number of ways to go about it. The limiting factor will be your budget.

Here are some options for you to explore:

Option 1: Wi-Fi-Based Multi-Room Audio System

The most reliable way to do this is with a multi-room Wi-Fi speaker system, like:

• Sonos – You can have speakers in each room (including outdoor-rated options) and use the Sonos app to group rooms together and stream in perfect sync. The bathroom can be handled by a waterproof Sonos Roam or Move, or you could keep using your Bluetooth mirror and handle it separately (more on that later).
• AirPlay 2 – If you’re streaming from an iPhone, iPad, or Mac, AirPlay 2-compatible speakers (like HomePods, some Sonos models, or other AirPlay 2-enabled brands like Denon, Bose, etc.) let you stream to multiple speakers at once across different zones.
• Amazon Echo or Google Home ecosystem – If you're in one of those ecosystems, you can group speakers and stream synced audio as well.

Option 2: Mixing in the Bluetooth Mirror

Bluetooth’s limitation is that it can only pair to one device at a time, and doesn’t support multi-room syncing by itself. So for the Bluetooth mirror in the bathroom, you can do one of the following:

• Run the bathroom audio separately – Just connect it to an iPad or spare phone streaming the same playlist or internet radio station as the other rooms. It won’t be 100% in sync, but close enough for ambient music.
• Use a Bluetooth transmitter/splitter – If you're set on trying to include the mirror in a wider system, a Bluetooth splitter that supports multiple devices (like Avantree Oasis Plus) might work, but it's a bit finicky and sync won't be perfect.

Honestly, I would skip option 2 all together.