Just working within the terms of the OP. And admittedly this is kinda marginal for GQ.
SC law may lay out the specific drugs to be used. If so, the state is stuck, but has been stuck for some time for all the reasons outlined above.
If not then IMO the state would be free to use whatever they want. You could (eventually) kill someone by infusing them with a fully saturated solution of distilled water and table salt. Or a lot of almost any other noxious chemical.
To be sure the state might be subject to widespread condemnation for using pine-sol or bleach on the poor inmate. And somebody somewhere might try to get an injunction against this plan. But if they a) keep the details a secret and b) can find somebody to do the deed, they may well succeed in their goal of killing the inmate and be left to deal with the legal fallout later.
Heck, if the law says “inmate’s choice, and absent that it’ll be lethal injection”, then what happens if the inmate does choose lethal injection, knowing the state doesn’t have and can’t get the drugs?
I could imagine a court, if asked, saying that although the law says the choice exists, when neither the inmate nor the state can implement choice A, it would be legal error contrary to legislative intent that the execution is thereby forestalled forever. As such, choice B is what the law demands.
Finally, there is nothing to prevent the legislature from amending the execution statute in light of changed circumstances ref the current practical impossibility of lethal injection. And once having altered the law, perhaps to require electrocution in all cases or to add the option(s) of hanging, firing squad, etc., the state could ahead and apply the amended law to the inmate.
IMO because this legislation would not be altering the crime of which convicted, not the terms of sentence, which is “death”, making these changes would not be deemed ex post facto.
YMMV. A messy business for sure with contradictions galore at its core.