Pretty sure the databook stats that the poison is 'heavy metal based' meaning it works through necrosis and entering the cells. Eventually, the cells would die and that would ultimately be the death of said person. Prior to that, the poison is sitting in the cells, and not killing them. That means there's plenty of time for Tsunade to accumulate the level of toxicity in her body. And yes, Byakugo only works when she has injuries. ISW is the first injury, and the cuts she receives is constituting many millions of cells which have a good chance of being split from poisoned cells. How hard is it for Sasori to cut her up again? Not hard at all actually, assuming he chooses to use his large supply of Iron Sand.
You must be registered for see images
Sasori chooses to make that the battle field and Sakura/Chiyo will have to watch out for attacks from below and from in front for every second of this fight. Not to mention IC Sakura will be tending to Chiyo's wounds with no antidote, more than enough time to strike her down then.
Pretty confident it's in the databook as well as the anime.