Hello aev_84,
Edit the Desktop: Default Profile. Click on the Assignments icon on the left sidebar. Drag & Drop the key assignment (s) to each button on the pictured mouse. You can create many different profiles for specific games but the Desktop: Default is the one you want to edit for everyday macOS use.

Welcome to the Apple Support Communities. I understand a Bluetooth mouse is not staying reliably connected. I'll do what I can to help.
- After updating to MacOS 10.15 Catalina, my Logitech Unifying devices (a MX Anywhere 2 mouse and a K750 keyboard) both stopped functioning. I can use both in the initial login screen, but once I'm logged in the keyboard stops working and the mouse mostly stops working - I can't move the pointer or use the first or second buttons, but scroll.
- The mouse is M325 and I have tried it with 2 different Unifying receivers. I am running Logitech Unifying Software 1.3.375 and Logitech Control Center 3.9.8 I have updated firmware on both Unifying receivers and after doing so the mouse never works on the receiver running v24.11(36).
- Standard OS driver works fine. Logitech Control Centre software is mainly needed to map the 'extra' buttons and features on some mice to useful functions. I have a MX Master, don't have Logitech control centre installed, and basic mousing, left/right buttons, the fancy scroll wheel and the horizontal scroll wheel all work fine.

Here are some suggestions:
Since you've already removed the device and added it again, try checking for any potential sources of interference: Resolve Wi-Fi and Bluetooth issues caused by wireless interference
Disconnect any external devices from your computer such as any external drives and then reset the SMC and NVRAM:

Logitech Mouse Catalina Island
If you are connected to a 2.4Ghz wireless network, try connecting to a 5Ghz network instead.
Also try disconnect from Wi-Fi altogether and see if you still have any issues.
Have a great day.
Logitech Mouse Catalina Software
Nov 13, 2019 9:25 AM