how can I link the filename to the visible layer?

Perhaps you can solve problem by using Window > Layer Comps but that option is requiring additional work to create layer comps although and that can be automated what depends on what exactly you need and whether group names are always the same.

If provided script works for you then you do not have many reasons to think about layer comps and File > Export > Layer Comps to Files.

@Сергій Марищук 

Are any of the replies so far helpful/correct?

with(activeDocument) saveAs(File(path + '/' + name
.replace(/(?=\.\w{3}$)/, '_' + activeLayer.name)))

I am using "Actions" to apply patches add badges to the posters.