Is it permissible to seek the help of apostates to claim rights from others, knowing that resorting to disbelievers or man-made courts is an act of disbelief?

Secularism, which excludes God's law from people's affairs, clashes with God's will and حكم. Resolutions of Fiqh councils and fatwas of scholars have stipulated that it is major disbelief (kufr akbar). However, this does not necessitate declaring specific individuals as disbelievers without observing the Sharia-compliant guidelines, as there might be ambiguities or ignorance that prevent declaring a person a disbeliever.

Regarding your question, there is no sin for a Muslim in resorting to man-made courts out of necessity, to reclaim their usurped right if they cannot access it otherwise.