If you were working for BP on a Liberian flag tanker, I bet that your employment agreement would specify that any claims must be arbitrated in London under U.K. Law. US law generally favors arbitration and choice of law provisions in contracts.
If there were no employment agreement, which MLC and U.K. laws both require, I expect the case would end up in UK courts.
If for some bizarre theoretical reason, UK courts would not hear the case, then it would fall to Liberia. If you could argue successfully that Liberia has inadequate corrupt courts and that you could not get fair consideration of your case in Liberia, then you could argue that since the Liberian Registry is based in the US, and BP is registered to do business in the US and has major offices and business operations in the US (sufficient contacts to confer personal jurisdiction) that a US court should hear your case.
Practically speaking, for reasons of financing, taxation, and limitation of liability, many shipowners have each ship owned by a separate corporation, and then have a separate crewing agency that employs the crew. Many US companies routinely do this, including Seattle companies and bayou companies. So that “BP tanker” you were working on may actually belong to an Isle of Man PLC, that is owned by a Ukrainian Trust, and you may actually be employed by a crewing agency in Malta. There will be some sort of MLC compliant employment agreement that specifies how legal claims are to be handled. It may not be fair.