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

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 )

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.

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