Best weapon for a bladesinger is a quarter staff or club?

For discussion about 9th level spells I would say Time Stop is best for Bladesinger. You can use it and then spend a few turns buffing yourself and activating things like your bladesong and shillelagh. Mirror Image, Haste, Blink, Crown of Stars - whatever you can get for however many extra turns you have. Sure, you'd only get them for 7 turns or so, but most battles don't last that long anyway and you're going to be very suddenly strong. You can save yourself one of those limited extra turns by using Contingency with the condition of "when I cast Time Stop". You can have Haste and Blink be Signature Spells to reduce spell slot use (more to use for defensive features/spells) and Mirror Image can be a Mastered spell to cast infinitely without spell slots so you can re-up it whenever you want (and still get to make an attack thanks to Haste, which can still be a cantrip thanks to special feature of extra attack). 

To the enemy you basically went Super Saiyan.