This part of the building complex (the one with the loggetta on the west side) seems to be a second part of the Archivio Arcivescovile, corresponding to the one on the north side of the Palazzo.

Please see the Wikimedia photo with my pop-up annotations:

https://commons.wikimedia.org/wiki/File:55100_Lucca,_Province_of_Lucca,_Italy_-_panoramio_-_jim_walton_(9).jpg