What happens if I refuse to do Granny Rags’ quest where she asks to poison the elixir supply? (spoilers)

In general, the optional missions that you should not complete are fairly obvious.

Specifically, you are correct, and the Granny Rags mission does increase Chaos. The Outsider is not specifically determined to keep it at a minimum, so his endorsement does not help in this regard. The Bottle Gang is helping to keep the plague at a minimum by selling their cheaper version of the Elixir; indeed, in a later mission you find a note that seems to indicate that even the guards use it if something happens to their ration. After you taint their stash, you spread the rat plague until they figure out what's going on, and the source of cheap Elixir is gone.

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)