Razer Basilisk V3 Pro: Bluetooth pairable with multiple devices simultaneously?

Good day,

I recently switched from Logitech (G900) to a Razer mouse, because Razer still offers Mice with 2.4GHz and Bluetooth, especially for gaming.

I was looking for a good gaming mouse with 2.4 GHz and Bluetooth. Shortly before I even had a "Logitech MX Master 3S" here. And if my mouse didn't have to be designed for gaming, it would be right back here! The "Logitech MX Master 3S" had a super great and easy to use feature: Via Bluetooth you can easily store 3 devices and decide via a button on the back between device 1,2 or 3. Is this also possible with my Razermouse?
At least I have a button on the back called "Profile". But i probably shouldn't get my hopes up

What I should also mention and which probably already answered my question itself, is when I have a device connected to the mouse, the mouse is no longer displayed on other devices via Bluetooth search. Normal. But then with the "Logitech MX Master 3S" I was able to press the button under the mouse to switch between profile 1,2 o. 3 and for each of the 3 profiles the mouse was also found again in the results for pairing. I just doubt that the "profile" button is for this.

If a solution is not implementable via bluetooth, is there a way to achieve it via a hyperspeed usb dongle? Because I have to the mouse currently the Mouse Dock Pro, with which the Hyperspeed usb dongle is currently stored only in the mouse. Could I plug that into my laptop and somehow switch between the dongle and the dongle in the Mouse Dock Pro?

I would like to be able to connect the mouse to 3 devices and switch quickly:

Main PC (Mouse Dock Pro)
SteamDeck (via Bluetooth)
Laptop (again with bluetooth or via the hyperspeed usb dongle not needed so far)

If all does not work, my only option is to unpair device every time I switch between SteamDeck and laptop > reset mouse bluetooth settings (because mouse currently still thinks it is paired after unpairing) > and just then other devices can find mouse to pair again.