It really depends on the Joint Operating Agreement that was signed by all parties. On most of the JOA's that I have seen, the operator is required to use standard industry wide procedures in their operations. If the non operating partners can prove that BP veered from standard procedures or was reckless in their operations, then they could get out of paying their share of the cleanup costs.
Below is an excerpt from a standard Joint Operating Agreement that I have seen in the past.
Designation and Responsibilities of Operator
(Company name here) shall be the Operator of the Contract Area, and shall conduct and direct and have full control of all operations on the Contract Area as permitted and required by, and within the limits of this agreement. It shall conduct all such operations in a good and workmanlike manner, but it shall have no liability as Operator to the other parties for losses sustained or liabilities incurred, except such as may result from gross negligence or willful misconduct.