After Agent Hunter escaped from "Real SHIELD", they have no idea about what he would do or where he would go next. It was safe to assume that after finding out that Bobbi and Mac were double-agents that he would want to warn his friends and colleagues in Coulson's team, and as such "Real SHIELD" realised that they might not have much time before their cover was blown. This spurred them into action.
Even in the event of "Real SHIELD" successfully taking Coulson's SHIELD by surprise, there is a very real chance that there are various protocols in place to protect, or even destroy, Furys toolbox. For all "Real SHIELD" know, as soon as an internal alarm is tripped, the box is automatically fired into the sun. Even if that isn't the case, it is likely that one of the expert agents on Coulson's team (such as Agent May) would make a beeline for Fury's toolbox once they realised what is going on.
As such, it was decided that they would use Bobbi's cover as a member of Coulson's team to attempt to secure Fury's toolbox before they attacked, which provided "Real SHIELD" with the greatest possible chance of successfully completing their mission before their cover was blown.