Yet another update....
I came across that discussion: Building debug libmbed.a
I can't follow all the details there, but I did give this alternative procedure a chance that is described there (post #28).
It worked...
It is also possible to specify GIGA:GIGA as configuration, which was not possible before for me.
Within the subdirectory variants/GIGA/libs/ I'm finding libmbed.a, which I'll try to move into the IDE somehow, resp. replace the original one.