Large cargo shortages or other substantial cargo-related claims are settled with the cargo insurers. Insurers first indemnify the cargo-owners for the damage and then, by way of subrogation, sue the carrier in the name of the cargo-owners if, of course, there is an actionable fault of the shipowner.
Per Thomas J in Protank Shipping Inc v TOTAL Transport Corporation (The Protank Orinoco), 2 Lloyd’s Rep 42:
If the meaning of "independent" is satisfied by showing that the surveyor in question is a firm or organization that operates independently of the owner or charterer or receiver, then there is always the likelihood that at any discharge port there will be more than one surveyor who is "independent" on this definition. If each reached a differing conclusion, then each could not be final and binding as between the parties. It follows, therefore, that the addition in the clause used in this charter-party of the words "whose estimate shall be final and binding" must have envisaged in the context of day to day discharge operations from tankers the appointment of a surveyor jointly on behalf of the parties to make a determination that was final and binding. Quite apart from that, I consider that it is highly unlikely that the parties would have intended to entrust such an important determination which would involve final decisions on significant sums of money unless the person was jointly appointed. In the standard form of the clause, a determination by an independent surveyor would not be final or binding; it would merely entitle the charterer to make a deduction from freight. This would have an adverse cashflow consequence to the owner, but leave the owner with his right intact to recover the unpaid freight, if he could establish that the oil was not pumpable. In the clause used in this charterparty, the decision of the surveyor was not as to cashflow but as to the final determination of the rights of the parties.
Per Moore-Bick J in Triton Navigation Ltd v VITOL SA (The Nikmary),  EWHC 46 (Comm):
28. In the present case the first inspection was carried out by Mr. Maiti, the charterers’ surveyor, alone. When Mr. Maiti carried out the second inspection he was accompanied by an inspector acting for the owners and on the third occasion an inspector appointed by the vessel’s P. & I. Club was also in attendance. On that occasion all three surveyors agreed that the vessel was clean enough to load. Mr. Hill was therefore able to submit with some justification that there had been a joint inspection and that the requirements of the clause had been satisfied in spirit, if not in form, even though none of the inspectors was acting on the joint instructions of the owners and charterers. The difficulty with that argument, however, is that the clause as a whole provides quite unambiguously for the vessel’s condition to be judged by an inspector appointed by both parties jointly and cannot be construed as providing in the alternative for a joint inspection of the kind that took place in this case.