The summary of the clarifications recorded in Biography, Hadith, Fiqh and Fatawa books regarding the clothes is that the holy Prophet Muhammad (صلى الله عليه وسلم) usually used to wear topi (cap) and amamah (turban) on his head like qamis and izar. He did not leave his head uncovered. In this case, most probably his must have the habit of covering his head during eating, drinking, and going to toilet. And when it is confirmed, there is no doubt being it sunnah.
A person who claims his habit of bare-headedness during eating, drinking, and going to toilet or he favours being bareheaded as sunnah it is necessary for him produce the proof from a hadith for his support. And whatever hadith he narrates write it with reference and then in-sha-Allah you shall be answered in detail.
Allah (Subhana Wa Ta’ala) knows Best
Darul Uloom Deoband