A anand B should consider a binding determination from a trusted neutral expert

As others have pointed out, this is a lesson in the importance of negotiating authorship order and work responsibilities prior to commencing a research project. Since that ship has sailed, as a suggestion for forward progress, if A and B cannot agree on authorship order (as appears to be the case), they should instead attempt to agree on a process for binding determination of authorship to which they are both willing to be held. It is sometimes the case that parties who are in an insoluble dispute over an issue can nonetheless agree on a reasonable process for binding determination by a neutral third-party.

One sensible option would be to negotiate for a trusted neutral expert (e.g., a senior academic in their field but who is a neutral arbiter on the matter) to review the history of the research agreement and the respective contributions of the parties and make a binding determination on the proper authorship order. Both A and B would have an opportunity to present their "case" to this decision-maker and would agree in advance to be bound by the outcome of the determination. This would allow them to "agree to disagree" but still push forward with the publication under an agreed process for determination of authorship.