It's not the job of itemization to fix balance problems between weapon styles, and overpumping this would be a big mistake. Ideally, each weapon style - and weapon - is matched in DPS or utility or some niche way versus each other. If you are worried that a Quarterstaff with appropriate dice (and the bonus I'll regret adding of a bonus crit mult) is not performing well versus a greatsword, we should buff Quarterstaves as a whole, not buff THIS quarterstaff to compensate for a class/weapon style/combat technique imbalance.
If I go 20000% on this and give it some absurd crit something, and in the future we change Quarterstaves by changing Thief Acrobat or Henshin or add a universal pole-vaulting tree, then we'll have to go back and firmly smack the Staff of Shadows with 14 individual nerfs and it'll be a whole thing and they'll be 20 pages of discussion about how we hate quarterstaves and pole-vaulting and the Olympics.