Start the mission as Bentley, then head over to the battery. It needs to be charged, and Bentley has an idea on how. You will be given control of Carmelita.
As Carmelita, keep shooting Sly until his health drops to zero. After Carmelita reports to INTERPOL, you will be given control of Sly.
As Sly, head to a mainline outlet to stabilize the battery. Watch out for Carmelita, though, the battery won't be able to take Carmelita's shots from her pistol at this point. Have the battery stabilize, take out any enemies during the process, pick up the battery, and head back to the Safe House. Plug it into the Safe House outlet, deal with any enemies during power transfer, and it's Job Complete.