I see now that I wasn't able to find this solution myself as I didn't know the name of the corresponding property (or what is it called?) in the lyrics context.
I. e. I didn't know what the analogue of "Staff.TimeSignature.font-name" for a lyrics context is called. Is there an overview somewhere that shows all property names of the different contexts?
Let’s say you’re looking in the Notation guide for lyrics stuff, e.g. <http://lilypond.org/doc/v2.19/Documentation/notation/techniques-specific-to-lyrics>.
At the end of each section/subsection, there is usually a “See also” section, which usually includes an "Internals Reference” link — in this case, there is a link to <http://lilypond.org/doc/v2.19/Documentation/internals/lyrics>.
There, you will find a whole whack of information about the settings and parameters for the Lyrics context. (n.b. In the left-hand navigation column, you can also see all the other contexts.) One important thing you learn is that “this context creates the following layout object(s)”, which in this case includes LyricText <http://lilypond.org/doc/v2.19/Documentation/internals/lyrictext>.
There you find all of the properties and settings, either “locally” (i.e., set here), or available through “interfaces”, listed at the bottom. THERE, you will find that LyricText inherits properties from the font-interface <http://lilypond.org/doc/v2.19/Documentation/internals/font_002dinterface>.
Here — finally — you find the user-settable property called “font-name”. Which is what we can set, as I demonstrated.
Essentially the same chain of links can be followed from any Notation guide page dealing with time signatures (e.g., <http://lilypond.org/doc/v2.19/Documentation/notation/displaying-rhythms#time-signature>), leading to the analogue Staff.TimeSignature.font-name.
This is precisely what I was looking for - thanks a lot.