The Chapter House used to be the venue for disciplinary congregations, business transactions with outsiders and the like, which does not really fit the bill of the alternative history threads.

I was thinking of another possibility: Claustrum, the Latin term for the cloister or half-open colonnade running along the inside of the monastic quadrangle. Open to the fresh air as well as to the vagaries of the mind, the cloister would be the ideal spot for monks to repair between duties and discuss casual topics, exchange gossip or address pressing matters of conscience and theological speculation in a less formal atmosphere.