I suspect there's a chance someone stumbles upon this thread with the same problem, so here's an ultra inconvenient work around that worked for me, so there's a chance it might work for you as well.
(Please proceed with caution, whenever my instructions include editing or replacing files you should make sure to keep a backup of the original file in case something goes wrong)
- Navigate to the installation directory of your Logitech Gaming Software (usually 'C:\Program Files\Logitech Gaming Software')
- Go to 'Resources\G403W\Manifest' if you have the wired version or replace 'G403W' with 'G403WL' for the wireless version of the mouse.
- Copy the 'Device_Manifest.xml' to a different location to be able to edit it with a text editor of your choice
- Use ctrl+f to search for 'force_swap="LeftRightMouse"' and replace the 'locked="true"' right after that with a 'locked="false"' for both instances in the file and save it
- Make sure to fully exit your Logitech Gaming Software before replacing the original file with the one you just edited. You'll probably need administrator privileges to do so
- When you start the Logitech Gaming Software again there's a chance you're good to go now, but if you're unlucky like me you will now find that the context menu when clicking the left or right mouse button will present you with the desired option to assign different functionality to them, but whenever you want to change it it will tell you that 'there needs to be at least one button assigned to left/right click, please assign another button to left/right click before doing this' despite having left/right mouse bound to different buttons already. So in order to get around this:
- Prepare the macros/functions/whatever you want to assign to left/right mouse in the Logitech Gaming Software by clicking the + Symbol and using the 'Add' button once you're done. Make sure to remember the names you give your commands. Make sure you have a button assigned to G-Shift if you wish to assign G-Shift+Left/Right mouse button to something else.
- Export the profile you're using by right clicking its icon and selecting 'Export...' in the Logitech Gaming Software
- Edit the exported file with a text editor of your choice, search (ctrl+f) for '<assignments devicecategory="Logitech.Gaming.Mouse.G403">'
- Beneath you'll find all your button assignments for buttons 1-6 including another instance of 1-6 representing the function they serve while G-Shift is active. Button1 means left mouse button, while Button2 means right mouse button. The 'shiftstate' in the end refers to if G-Shift is held or not. Depending on how many Logitech G devices you have/had there might be differences in what number means what, but 'shiftstate="1"' should always mean no shift button is held. For me the shiftstate for the G-Shift button on my mouse was 'shiftstate="4"', so whichever number you find there that is not 1 should be correct. Maybe it's just always 4, I'm not sure.
- Next you'll need to find the 'guid' of the functions you want to assign to the buttons, use ctrl+f and type in the name you gave your command to find it and copy its 'guid' (it should look something like this: {DH7302E2-EB5E-4A08-AE33-5217D6FGC7D6})
- Copy the 'guid' from there to replace the 'macroguid' of whichever button and shiftstate you want to change (Since you're editing all this manually you need to make absolutely sure that you end up having at least one button assigned to left click and one to right click, otherwise you might get into trouble reverting your changes)
- When you're done, I'd recommend you to change your profile's 'name' in the very top of the file to something different so you can tell the two apart easily. Save your file
- Now you should be ready to import your manually edited profile back into the Logitech Gaming Software by clicking the small arrow down right under the folder icon and choosing 'Import...'
- Switch to your newly imported profile and see if your button assignment ended up the way you wanted. As long as the button-gshift-combination you want to change is not currently assigned to be left/right mouse click you should be able to change it within the software now. Just make sure not to assign left/right click to anything else as that might very well get them stuck with that function again, requiring you to go through the manual editing once again.
I would like to stress that I do not consider this issue fully resolved and that I'd hope Logitech can come up with a better solution, as this workaround is pretty inconvenient and definitely shouldn't be necessary to use the mouse in its full functionality as advertised.