Found an almost perfect solution, with thanks to @macydanim to pointing me in the right direction.
In Natty, Banshee won't show up in the notification area, so I ran the following command:
gsettings set com.canonical.Unity.Panel systray-whitelist "['JavaEmbeddedFrame', 'Mumble', 'Wine', 'Skype', 'hp-systray', 'banshee']"
This adds Banshee to a "whitelist" of applications allowed to be shown in the notification area.
Logged out, and logged back in again. Then in Banshee, went to Edit -> Preferences -> Extensions. Disabled
Sound Menu Integration and enabled
Notification Area Icon. Then, right clicked the notification icon and disabled show notifications.
I then disabled
Notification Area Icon extension, and tried re-enabling
Sound Menu Integration. This, however, still showed notifications whenever the track was changed. I found out though, that you don't need the
Sound Menu Integration plug in enabled for Banshee to work in the sound menu. So if I leave it disabled I don't get notified on track change, but can still operate Banshee from the sound menu. The only downside, is that I can't click the close button on Banshee and have it minimise to tray, so you just have to remember to press the minimise button now instead of the close one and keep Banshee open.