It's a bit poor that with today's technology they can't override the damned thing. I'd have thought if they blanked out the dx coding on the cartridge it might ask for a manual input. never mind.
All C-41 films take the same processing time, at least in theory. Thus, there's normally no need to adjust the time for different films or different film speeds. AFAIK, if minilabs read DX codes at all, it's not to tweak the development time. Covering the DX codes might seem a clever way to signal that a roll needs special treatment for push or pull processing, but the cartridges used inside disposable cameras usually lack DX codes, IIRC, so this would just slow down the process, perhaps confuse clerks, and maybe even result in errors if the confused clerks enter the wrong data. That's not to say that an override wouldn't be useful, but covering over DX codes wouldn't be a good way to activate it.
Now I'm torn between pushing Tri-X, which I've done and is okay, or shooting Neopan 1600 to continue to take advantage of cheap minilab processing.
Fuji Neopan 1600 is a conventional (non-C41) B&W film. As such, drug store minilabs either won't process it or will return blank negatives. Trust me; I once made the mistake, before I began doing my own processing, of taking a roll of conventional B&W film to a local CVS. They ran it through their on-site minilab machine, handed me back a blank negative strip, and had the nerve to suggest that my camera was at fault.
That said, pro labs will process conventional B&W films. I don't know if minilab machines can be used for this purpose or if they all do it in manual tanks or whatnot. The only remotely pro lab near me will do B&W. I'm not positive, but I think they do the film manually and then make 4x6 proof prints from the negatives on their minilab machine.