Given the name "post-battle" I think one would be hard pressed to argue this is within the duration of a single battle
RAW there are various reasons why it can't be done, I think we all agree.
But this one is a bit weird. Why can't you take your drugs unless you're in a battle? Do you reach into your jacket pocket, take out the syringe, roll up your sleeve and... the needle won't go in? Does the Escher Police appear and say, 'Stop! Life-saving drugs are for gang-fights only, not for medical or recreational use!'?
Anyway, it's just an idea.
Edit: Here's another idea. How about selling doses of Night-Night to other players?
I wouldn't sell Hyper to a Corpse-Grinder gang, personally, but I wouldn't mind selling them a few doses of Night-Night out of my stash. Buy them for 10 credits thanks to my Chem-Alchemist, sell them for 30, and the other gangs get the benefit of knowing that their Leader won't die horribly in the next battle. It helps with my economy, which lets me afford even more drugs, and then everyone can do the Mnemonic Spike trick if they feel so inclined.
It also lets me choose who I sell to and who I don't, so they had all better be nice to me! Maybe refuse to sell to Enforcers. And Goliaths. It would be up to me, really.
Ha!