BEIJING—The landlocked African kingdom of Lesotho doesn’t have an obvious stake in the South China Sea, but it is among some 60 countries that China says stand behind it as it faces potential censure by an international tribunal over its territorial claims there.

The sudden involvement of Lesotho and other small nations far from Asia is the product of a Chinese blitz to rally support in the final countdown to a ruling in The Hague, which could come this month, on a case brought against China by the Philippines.