QGIS Advanced labeling features with atlas

Not an answer on the desired expression - I'm pretty sure it would require some scripting rather than expression editing. But here's some advice for label decluttering:

• For point features perhaps generating callouts is an option to avoid overlapping

• For polygons use free placement (you did that)

• For line features depending on complexity consider callouts as well

• Symbology tweaking:

Generally you should always rethink the label's information: Is there a way to implement a label's information into the feature's symbology? This applies especially for multipolygons/multilines. For example you have pipeline A labeled 'A' and it has some branches labeled 'A subbranch 1', 'A bypass 2' and so on it would be efficient to add a rule based symbology on the piplines and make the legend do the labeling

• Attribute table with abreviated labels:

If symbology tweaking is not that much of an option for you, be it that the labels are all different/unique you can still generate a table like Ian Turton already commented: label your features with 1,2,3 a,b,c and so on and add the table (1: name of feature labeled '1') to the layout.

• Overlapping lables:

Last but not least you could try enabeling overlapping labels for some layers and tweak the opacity and blending options for fill and outline for these overlapping labels. This might require some tinkering to eventually look satisfactory but I personally had good experiences with this technique.