I have a Logitech G502 and use the side button for sprint (and another ability with Shift*, Alt*, Ctrl*) and notice it did not work on PS5.

I found that you can use the Logitech software on your PC to make side button function as another key on your keyboard. And save the setting to the on board memory on the mouse. I used the P key (something I do not use on the keyboard) And I programed that as the side mouse button. So when press the side button. The PS5 thinks I'm pressing the P key.

So that is one way to map your side buttons on your mouse for you PS5.