I assume the 6-digit code subdivides the 4-digit code, but no 6-digit code should be in more than one 4-digit code. I would start by using the Dissolve Tool on the Building polygons with the 6-digit postcode and the 4-digit postcode as two fields in the unique case portion of the tool. I would check to see if my assumption was correct by sorting on the 6-digit code and looking for any instance of more than 1 polygon containing that code. I would also use the Summary Statistics tool with the 6-digit code to find the same thing.
The Building polygons will have gaps between them. They also should be contained within a single 4-digit polygon. To get the best boundary I would manually divide a copy of the 4-digit polygons by using the split polygon tool and tracing the outer shape of the dissolved buildings and using centerlines as a snapping feature. The Spatial Join tool could then be used to the original building polygons to append the 6-digit postcode to these polygons, with the manually split postcode polygons as the target features and the buildings as the Join features.
I could also try the Aggregate Polygons tool, but that tool will form boundaries with gaps and overlaps between adjoining 6-digit polygons and bleed into adjacent 4-digit polygons. I would use the Union tool on the resulting 6-digit aggregated polygons with the 4-digit polygons and throw away any portions where the 6-digit polygon fell into the wrong 4-digit polygon and then redissolve the results. The overlaps between the 6-digit polygons inside a 4-digit polygon would be fixed manually to get the best result. While forming centerline outlines into polygons could help to contain some of the building groups, other blocks will be split between postcodes and still require manual intervention where the 6-digit polygons overlap.
Anyway, aggregating divided polygons is easy and can be automated. However, subdividing an aggregated polygon is hard and is more art than science, so it almost always involves a manual step to get the best results.
