Just a hunch, but I would guess that a poor disguise would be more suspicious than no disguise. If I was a shopkeeper and government agents came to me asking if I remembered anyone suspicious in the past month, I'd probably recall the guy with the obviously fake mustache and prosthetics.
On the other hand, an effective disguise can be trivial: enter a store dressed a certain way, make purchase, exit dressed significantly different. Reversible jacket, headgear swap and glasses can go a long way, in a crowd under low-res cameras.