Regiongroup first, is what I normally do first, if there is any chance of duplication of zones (akin to multipart to singlepart in vector). I don't like multipart anything and rasters do this by default. So if the 1km raster values repeat, I would instinctively regiongroup before I proceeded (if I was using SA and not arrays anyway.)
I suggested ZonalStatisticsAsTable for this reason the ambiguity issue is mentioned in the help files. Plus the multipart nature of a 'zone' can lead to different results for 'majority' and 'minority' since distinctly separate areas belong to the same zonal class may not exhibit similar properties because of the zone aggregation
....."or majority and minority calculations, when there is a tie, the output for the zone is based on the lowest of the tied values"...
I generally don't like zones or multipart anything unless you know what you are dealing with, but it may be a workflow preference