How can I change default mouse buttons on a non-mac mouse?

I've used USB Overdive for many years: it's effective and stable. There is a trial version you can use for as long as you like - it just adds delay before starting: make sure you get version 4.0.1 for Catalina (the latest version is Big Sur only). You can customize the buttons (I use it with a trackball with four buttons but you can have more) and have different actions for different applications: these include typing text with modifiers.

https://www.usboverdrive.com/

You would think the maker of the mouse would provide some software to handle it's nonstandard keys. It may only provide a windows driver.

What is the make, name and model number for the mouse.

You could use some third party remapper to change what f3 does. I use keyboard maestro to remap my function keys and other things.

R

I have no idea if this is a spam free website:

https://www.driver-free.com/download-bengoo-km-1-drivers-software-wireless-gaming-mouse/

This seem to be all window's drivers. I did look on the manufacturer's web site, but could not find driver.

Could try connecting to a windows box and programming the function keys how you like. see above.

R

ps. problem with no-name brand

Without the 64 bit drives for macOS Big Sur installed on the computer - IMHO - the programming of the mouse is going no where.

Although, their site USA is showing PC / Mac - the drivers appear to coming from a One Drive site and that again, IMHO, is as doggy as heck.

You migt want to try the USB Overdrive software. Its purpose is to offer additional control to third party devices like mice, so it might help. (Disclaimer: I do not have this software, and only used it a long time ago; I think there is a demo version you can try).

As already suggested - check with the Manufacturer of this mouse for Apple Supported Drivers specific for Catalina ( 64 Bit Driver only - 32 Bit drivers will not work )