The new facility to export CIF labels could be extended with options for a "label format string", in the spirit of specifying date formats as %d/%m/%y vs. %YYYY-%MM-%DD etc.
e.g. %L could refer to the CIF label, %I to the _space_group_symop_id.
At the moment if Z' > 1, a given CIF label is used multiple times. Specifying a label format of %L.%I would give labels e.g. H1.1, H1.2, H1.3 to distinguish all sites. This would remove any ambiguities in how sites with the same label are ordered and preserve the maximum information.
Atoms sitting on special positions would need special treatment. You could use a different % to specify "use the first symop" or "concatenate" e.g. H1.24 (Note sure whether the specification of space_group_symop_id as a char literally means single character).
Anonymous