If you want to use a Shield, you run into an issue though. The Quarterstaff can be used as an Arcane Focus for your Wizard spells. But spells picked up via Magic Initiate aren't added to your spell list, so the Quarterstaff cannot be used as an Arcane Focus to cast Shillelagh. Shillelagh has a...
I think your issue is with CME and in that case, you should just ban the spell. I don't see any min-maxing here. This is just basic optimization using CME.
Without a Vicious Weapon and going with the Shillelagh-build, the Eldritch Knight at level 11 gets (1d12+5+2+2d8)x3+2d6=74.5, doubled with...