I’ve seen it debated both ways on FR. Asylum can always be sought at an embassy. Can anyone point out cases where it was sought at a place that was neither an embassy nor a port of entry?
This does not directly answer your question, but the page linked below implies that U.S. consulates and other organizations can help refugees seeking asylum.
If you believe you need protection as a refugee, contact the United Nations High Commissioner for Refugees (UNHCR) or another international nonprofit volunteer agency. If these organizations are unavailable to you, contact the nearest U.S. Embassy or consulate.
at the detention center