Eilenberg's rational hierarchy of nonrational automata & languages

It would be hazardous to guess what Eilenberg had in mind, but a reasonable answer to your question

What material is likely closest to his ideas?

might be the theory of cones (also called full trios). An excellent reference on the part of this theory related to context-free languages is Berstel's book Transductions and Context-Free Languages, Teuber 1979.

An alternative answer (still for context-free languages) can be found in the little known but very inspiring article by J. Berstel and L. Boasson, Towards an algebraic theory of context-free languages, 1996, Fundamenta Informaticae, 25(3):217-239, 1996.

To quote your introduction, the first reference is using "rational relations as a tool for comparison" and the second one really "encounters algebraic phenomena which lead to the context-free grammars and context-free languages".

I recommend Behle, Krebs, and Reifferscheid's recent work on extending Eilenberg's fundamental theorem (that is, the correspondence between pseudovarieties of monoids and varieties of languages) to non-regular languages (link). They point out previous works in this line (in particular, Sakarovitch's on CFL).