The headword occurs only here. Its -a/rios suffix indicates it is patterned after Latin terms, such as kagkella/rios "chancellor" -- which originates in the same metonymy: cancellarius > cancelli "lattices, crossbars, which surrounded the seat of judgment". The word kagkella/rios is attested in Greek (Lampe s.v.), and kigkli/s (LSJ s.v.), being specific to courts and assemblyhouses, has the same associations: kigklida/rios may be a partial calque.
[1] kappa 1582, kappa 1583.
[2] Generally, town halls, boards of magistrates; here evidently, important records and paperwork, literally the archives associated with the military campaign; cf. LSJ s.v. a)rxei=on (web address 1).
[3] Agathias, Histories 1.19 (on Narses [nu 42] at Ravenna); glossing not the present headword but a more general category of officials. On the entourage with which Narses retired to Ravenna (rho 4) in the winter of 553 CE, see also omicroniota 79.