Use something like xbindkeys or sxhkd to bind mouse buttons to special commands/actions. This is totally out-of-scope here. I've a Logitech Performance MX, and simply use sxhkd to bind my extra mouse buttons to special events, there's no point involving steam in this (would you also want every non-steam game to implement support for your mouse as well?)
Closing as this is not Steam-related.
Logitech publishes their remapping software for Windows and OSX only.
Some hardware only functions properly with Logitech's software, and cannot do so in Linux, even with third party programs.
For example the MX518 has two DPI buttons that can be remapped in Windows and OSX using Logitech's SetPoint software, but not in Linux where SetPoint is not available (the problem is only with the newer version, dev C051, not the older dev C01E).
For the older MX518(C01E) there was a messy, ugly workaround using a third party app named Lomoco, but for the newer MX518(C051) not even that is an option - it is not recognised by Lomoco and there is absolutely no way to use the DPI buttons for anything other than DPI changes.
We need the same ability to use our Logitech hardware that Windows and OSX users have. The ideal solution would be Logitech publishing a Linux version of SetPoint.
For anyone wondering why I'm reporting this as an issue here, it's because for a similar situation with Razer mice, Mike Sart told another Linux Steam user to report the problem here, so I'm now doing the same for Logitech. https://github.com/ValveSoftware/steam-for-linux/issues/957