Based on the provided quotations, it is not possible to find material that directly addresses the symbolic meaning of prescriptions regarding head coverings for non-Jews in the synagogue or their role in religious tradition.

Insufficient Evidence on Synagogue Head Coverings

Based on the provided quotations, it is not possible to find material that directly addresses the symbolic meaning of prescriptions regarding head coverings for non-Jews in the synagogue or their role in religious tradition. Among all the cited sources, the focus is on the traditions of wearing head coverings in the context of other religious customs – for example, in Orthodox Church orders, where head coverings symbolize spiritual authority or service to the church hierarchy (for example, the quote from link txt, without specifying the page), as well as descriptions of the history of head covering development in various Christian traditions (for example, from link txt, pages 86-87 or from link txt). However, none of the excerpts specifically mention the head coverings of non-Jews in the synagogue or their symbolic significance in this context.

Therefore, unfortunately, the provided materials do not provide grounds to answer the question regarding the symbolic meaning of these prescriptions and their place in religious tradition specifically for non-Jews in the synagogue.

Supporting citation(s):
"Without an omophor, the archbishop does not perform any service, and in his attire, the archbishop represents Jesus Christ... Each sacred garment has symbolic meaning, reminding of the importance of serving the church hierarchy." (source: link txt)

"This custom also existed in the West... for covering the shorn spot – in Rus', it was called 'gumenets'..." (source: link txt, page: 86-87)

These examples demonstrate that the provided quotations pertain to the traditions of wearing head coverings in other religious contexts, but do not include discussions focused on the aspect addressed in the question.