It would require coding to restrict the buttons that display in the quick editor. Give me some time, and I will see if I can put together a product that will allow you to determine exactly which buttons appear on your quick editor.
I already have products available for adding buttons to the quick editor (and determining which of your custom BBCode buttons to add) and for adding information to the mouseover tooltips for your custom BBCode buttons:
Quick Editor Buttons [v1.5]
Editor Toolbar Tooltips [v1.1]