kind of solved: it appears I need QGIS for this task.
Joining attributes by the biggest overlapping areas in QGIS
• Largest overlap—The feature in the join features will be matched with the target feature with the largest overlap.
It appears I want the opposite: The feature in the Target Features will be matched with the Join Feature with the largest overlap.
Situation explanation
I've a Target-Features layer with many small polygons. And a Join Feature layer with larger polygon features. I would like to join the Name from the larger 'join features layer', to each smaller polygon in the Target Feature, but only the Name from the Join Features layer with the largest overlap with the Target Feature. What could happen is that a smaller polygon from the Target Feature layer, overlaps a bit with two polygons from the Join Feature layer. Then I want the name of the polygon feature with the largest overlap in the Join Features layer.
problem
Now I get only one join from the Join Features to the largest small polygon in the Target Feature layer probably because that one has the 'largest overlap'.
partly solution
In Fields, there is an option to Concatenate the name(s) of overlapping polygons from the Join Feature layer to one Target Feature.
It is not sure whether the overlap is 99% or 1%, but at least there is something.