Is it permissible to claim another illness as a cover-up to protect the privacy of the original illness, and what is the legitimate Sharia-compliant way to maintain secrecy without lying?

Avoiding lying is achieved by using tawriyah (allusory speech) and ma'arid (ambiguous statements), which means the speaker intends a meaning that the لفظ (wording) can bear, but which is contrary to the apparent meaning. Examples include the action of Prophet Ibrahim (peace be upon him), Abu Bakr's statement about the Prophet (peace be upon him): "He is a man who guides me to the way," and the Prophet's statement to the disbeliever: "We are from water." The Salaf (pious predecessors) used tawriyah to avoid lying.