Oddly enough the mousescroll function works when I'm in fullscreen, but now if I use match window-size but not the X-fullscreen button. Not tried putting yet as I can't enter any tournaments (will give practice a shot), but it works for changing clubs and scrolling up and down the leaderboard screens, so I will assume it works for changing the putter too.
Hope is not lost yet ;)
Edit: Yep. Works for the greens too. Just use fullscreen by hitting the bottom-right X and you can mousescroll to change your clubs and putters, the screens and so forth.