This addresses shop cameras, not street-level government surveillance that would be accessible for 10+ years and is trivially addressable by the time and location of the transaction.
Probably, but doing it correctly would require a whole different type of opsec expertise that should not realistically be expected from someone who is getting their security tips from The Intercept.
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.
This is the solution, right? You get someone else to buy the phone for you. This necessarily will practically speaking be probably be someone who knows you, but it adds a layer of indirection.
The unregulated pharmaceuticals distribution market relies on burner phones and seems pretty solid despite effectively continuous attempts to conduct surveillance on them.
Perhaps not something insanely obvious like Grouch Marx moustache and glasses, but more subtle? I'm not really very well aware of how advanced face recognition, etc. is at this point in time though. But I'd guess anyone who cares enough about opsec to go through this process might also be decent at passable disguise techniques.