Elementary bounds on parameter in fixed-parameter tractability?
In his dissertation thesis "Modiﬁzierte parametrische Komplexitatstheorie", Mark Weyer considered, among other things, hierarchies within FPT w.r.t. the function f and reductions between them.   He also did indeed relate these sub-hierarchies to fragments of FO and MSO: Chapter 6 is essentially about the relation between FO/MSO (the number of quantifier alternations of the formulas) and the function f(w) in Courcelle's theorem (w being the treewidth).  He considered both upper and lower bounds and, using the aforementioned reduction framework between certain hierarchies within FPT, he was able to give quite tight bounds.  The examiners of the thesis were Flum and Grohe.

Unfortunately the thesis is in German, and I'm not aware whether material of his thesis has been published in an English journal.  I'm therefore aware that they might be of limited use for you, but nevertheless the references therein might be a good starting point.