[ I've attempted to tidy up your abomination of quoting ]So use 10mm² 3-core SWA to the outbuilding. Then you've got a choice of using one of the cores or the armour, or both. Since, as is often observed, the difference between an earthing connection and a bonding one is just naming, there's no reason why a conductor cannot have two names at the same time.As far as the installation is concerned, the earth electrode at the outbuilding counts as an extraneous-c-p - which means that, with TN-C-S, a bonding conductor with a CSA of at least 10mm² would have to be run to the outbuilding if the TN-C-S earth were being 'exported' to there.
Sure, but (unless ≥10mm² is needed anyway), that only makes sense if there are other ('conventional') extraneous-c-ps entering the outbuilding. If not, it surely makes more sense to not export the TN-C-S earth but, instead, to simply rely on the local ('TT') earth electrode (now connected just by a 'earthing conductor', which is not also a 'bonding' one!) at the outbuilding?
Kind Regards, John