You could try the following:
Current (potential) workaround:
I do not know how these buttons are supposed to be named in config file.
So I would do the following:
In AutoHotkey:
#IfWinActive ahk_exe Obsidian.exe
XButton1::F13
XButton2::F14
#IfWinActive
Then you can assign F13 and F14 to Obsidian using the workaround described in Unable to set a hotkey to F1 - #3 by malecjan (post #3).
Note: I do not have multibutton mouse to test with so I actually remapped another key to F13 and it worked. Let me now if it works also with real mouse buttons.