If US society is so squeamish over the exact method of execution, it sounds to me that they're not really committed to the idea and probably shouldn't be doing it.
Arizona, for example, abandoned hangings after a noose accidentally decapitated a condemned woman in 1930.
Really? I thought the object of hanging was to break the victim's neck instead of strangling them, in which case this sounds like a resounding success - or did the baying crowds want to think that the victim had "just gone to sleep"?
I'm not a doctor, but I believe that pointing something called a "gun" at the subject and pulling the trigger a few times does the trick, is typically less unpleasant than many natural/accidental deaths and is already widely accepted as a method of dealing with foreign teenage conscripts who's governments threaten the stability of the oil market. Alternatively, every properly-run abattoir in the rest has a variety of efficient methods for killing large mammals with the minimum of suffering. If that's not acceptable then it suggests to me that you're not really 100% at ease with this whole "death penalty" thing and probably shouldn't be doing it.
Any society that wants the death penalty needs to be satisfied that "the end justifies the means" and understand that (a) there is no nice, guaranteed painless, dignified way to execute someone that won't go wrong from time to time and (b) you'll end up executing innocents occasionally. If you're not OK with that, don't do it.
Personally, if I were (rightly or wrongly) condemned to life imprisonment I'd like the coward's option - but not the USA version where, it seems, you rot in prison for a decade or two anyway and only then get dispatched by the sort of bizarre, theatrical method that a movie super-villain might dream up.