Possible to link mouse buttons to macOS functions
It's no doubt possible to map your mouse buttons to Keyboards keys, but considering the generic nature of your mouse, I'm doubtful you'd be able to do it without 3rd party software.

While I haven’t used these to map buttons to a HyperX Pulsefire FPS gaming mouse, I have used these to map the buttons from a variety of other controllers:


USB Overdrive 
ControllerMate


I’d start with USB Overdrive as it offers a much simpler interface. ControllerMate provides a lot more advanced functionality, but also takes quite some time to suss out.

Now, I haven't used SensibleSideButtons, but that may work in your scenario if the extra buttons are side buttons. Nor have I used Steer Mouse, but I know of people who swear by it.