What are the consquences of poisoning the Bottle Street Gang's elixir for the Gentleman Callers quest?
What are the consequences of poisoning the elixir?
You'd earn another rune from Granny Rags, and there's a bone charm in the distillery where you'll be poisoning the elixir supply . Also, there are some dialogue changes with NPCs, most notably Slackjaw (when you deal with him in a later mission).
Slackjaw and his gang sell a cheaper bootleg version of Sokolov's elixir. Slackjaw would mention that someone poisoned the elixir supply in the distillery, when you deal with him in a later mission. I also remember guards or an in-game note mentioning that the guards also use Slackjaw's elixir as it is cheaper (when you have not poisoned the elixir supply).
I believe its effects on chaos will depend on how you did that part of the quest (not whether you did that quest or not). Also, you risk higher chaos if you choose to do it.
You may also be curious on how this will affect the part of 'The Flooded District' mission...
..where you'll be meeting Slackjaw and Granny Rags again.
I have done two separate playthroughs: one where I have not poisoned the elixir supply, and another one where I poisoned it. I can say that it will not affect that part of 'The Flooded District' mission.
Is there another way to complete the quest?
There is no alternative method to complete that quest, AFAIK. The alternative is not to do the second part of the quest at all. Avoiding it will make getting the 'Ghost' and 'Shadow' achievements much easier, and has less high chaos risk (but will prevent you from getting the Gentlemen Caller and Street Conspiracy achievements, aside from the rune and bone charm previously mentioned). (Source)
This answer doesn't directly address the game mechanic but as far as the story goes, anyone worried about the moral implications may be interested in reading a note that can be found on a table in the second half the warehouse:
Recipe For Craxton
=================
Craxton,
I'm coming tomorrow to check on the batch. Make sure you're wearing pants this time, and stick to the recipe:
1 part Sokolov's elixir
1 part beechgum paste
2 parts sugar water
That's it. The more of the real shit you cut in, the less there is to spread around and the less coin I make. This ain't a charity.
Slackjaw
The inference being that the bootleg elixir wouldn't be effective against the plague anyway.