“Faking deaths is more making a big flash and erasing the paper trail.”
What that means is the important part of this plan isn’t insuring they appear dead to observers. No need for any drug to make them appear dead to medical staff which would be difficult and too risky (i.e. a neurotoxin might due the trick, but the difficulty would be delivering the toxin without being noticed and delivering a antidote before any permanent damage is done) Rather, making them seem dead is as simple as setting up a situation that could kill them (i.e. shooting them publicly, public executions always make a statement) and making them dead on paper.
Mostly likely Tseng shot them both with blanks. Blanks are not in fact harmless. When I asked her how much damage a blank could do she told me you could kill someone with a blank if you hit them at close enough range in the right spot. When I expressed doubt that blanks would be enough to knock them out flat, she insisted
“Blanks are less likely to cause permanent damage than bullet (read, think Watson's shoulder in Sherlock), but they will have knock back, and will likely cause a very nasty looking wound. Don’t believe me, do yourself a disservice and look up (with safe search off) "rubber bullet wounds" and "blank shot injuries"
I took her at her word for it and did not google images.
At the range Tseng shoots them he could serious damage, even with a blank. Certainly enough to make it look real, in order to avoid killing them Tseng would shoot them in the
shoulder. (I think the problem we were having is we were assuming a head shot, which wouldn't work, but the game doesn't show us where Tseng shoots them. Logistically, it would have to be in the shoulder). Although there is still the risk of damage and arties he could risk hitting (aorta/major brachial artery), it’s the best bet. Aim for the shoulder you won’t risk hitting organs, in case they aren’t wearing bullet-proof vests or some major armor equivalent, or revealing the gun to be loaded with blanks in case they are wearing something a blank won’t pierce.
Medical staff can help them later, because it’s best if medical staff are in on the scheme to insure we have solid paper trail, as mentioned earlier. With fake paper trails, it’s better to make a trail than fake whole cloth and having some helpful staff to make sure Elfe and Veld don’t die and to get them away would be useful.
As to the logistics of getting medical staff willing to aid in this deception, the name Rufus Shinra goes a long way. Help Rufus, and you’re guaranteed he’ll be a position to help you, refuse to help or threaten to betray his plans, you’ll be in a position that won’t be enviable. The fact Rufus can be dangerous would need to be clear, because betraying President Shinra is not a risk everyone would take for reward alone so Rufus must make the idea of not joining him just as risky.
Of course, we can’t know the exact details of what happened or how many allies were involved in the cover up. I’m also not entirely sure what role Reeve played in all of this, but I’m convinced he was in on something. However, the bulk of the work on the paper trail end would be on Rufus, and he’s the perfect one to do this sort of thing. He can access and alter whatever data he likes, and has influence. The paper trail, which would usually be the hard part in this sort of thing, is the easy part for Rufus.
All Tseng has to do is shoot them at close rang with blanks and hope they don’t die. Fortunately, Tseng has good aim so he could probably hit the least lethal spot. (Yes, it’s risky, but the Turks don’t have many options and it could work). If Tseng is quick he could discretely drug them both, or he can leave it to the medical staff that come to examine the “bodies” upon arrival.
“providing the shot wasn't to the head, the deaths would have been 'slow enough' to get someone to drug them to comatose stupor”
It would probably be better for Veld and Elfe’s comfort and for looking dead to any casual observers if they’re drugged out. Slipping them a potion and then drugging them might also work if he gets the chance, which Tseng might have had since specifically he rode in the back of the truck with the bodies.
“Any heavy sedative- the cocktail they use to induce comas would likely be the best. And given it's ShinRa, he will likely have access to their height-weight-allergies, thus able to organize the cocktail better”
I’m not sure of the likelihood, Veld’s medical records would be easy enough to get, Elfe’s data would require getting into Hojo’s records and who knows what state Elfe’s body is in post everything, but if drugging was necessary, it’s likely Rufus could arrange something or find someone who could help supplying something, possibly.