Your example seems a bit more like an example of a difference in interpretation rather than methodology, no? It seems like the issue with the French Revolution isn't whether we should say privilege diary sources over economic statistics or political documents, but whether the French Revolution was a success or a failure. Our methodology will influence our interpretation, but it seems that things outside of methodology might do so as well.

A problem you might have is that it might be hard to figure out who exactly is a conservative historian. To stay with the French Revolution example, this topic has been a battle ground between conservatives, liberals, and Marxists for over a century. Recently revisionism has become a major hallmark of the French Revolution (basically the rejection of the French Revolution as being a bourgeois revolution lead against the aristocracy) and from my readings on the topic this often involves a reevaluation of sources (generally rejecting the class structure that Marxism would have us see) and a focus on political documents and how those political documents and ideologies lead to the Terror. As far as I can tell, revisionism in France is a fairly conservative enterprise, yet in my own university the French Revolution people were mostly revisionists yet would be, in the US, political liberals (e.g. they voted for the Democrats and supported universal healthcare). So, since they embraced a particular style that is conservative, are they conservative historians despite being US liberals? Additionally, what do we make of feminist interpretations of the French Revolution which look at the gender ideology of the revolution, which is also revisionist in one sense, it rejects a Marxist interpretation of the revolution, but hardly seems conservative?

Another problem that one might have is that a particular position might be associated with one political ideology but over time dominant all sides. To give an example, in German history, the Sonderweg (the idea that Germany had a special path leading to the Holocaust) was embraced by many Marxist and liberal (Weberian) historians in a weird joint melding of both ideologies, but was generally rejected by more conservative historians who viewed the Hitler as an "industrial accident". Yet two young British left-wing historians effectively decimated the Sonderweg interpretation of German history (without embracing the industrial accident model) to the point where while there are still some major supporters, no one in my department supports it. (And this itself was a deeply methodological dispute leading to very different interpretations, but is just a bit hard to summarize. Basically I would read it as questioning whether "bourgeois revolution" was a useful category of analysis.)

Perhaps you might get more luck wondering whether conservative, liberal, and Marxist historians talk about methodological differences and they do, but often I am unsure whether these distinctions are really true. For example, historians like Crowley, Ferguson, etc. embrace counterfactual thought experiments in history and reject teleology in history, but counterfactual thought experiments have been embraced by those on the radical left (see "Agendas for Radical Historians" in Radical History Review) and anti-teleology is pretty pat these days. Perhaps one thing is the interest in "great man" history (see Crowley's What If? 2), which I am pretty sure most non-conservative historians reject outright.

It seems plausible that there are methodological differences between the two, but these might be subsumed in greater debates (e.g. how quantitative should history be) and additionally might change over time (e.g. it seems to me that the quantitative turn in history was primarily moved by left-wing historians, but these days that methodology is rarely used in history and is mostly the domain of economic history which is more liberal or conservative than left).

References:

For the French Revolution: Recent Debates and Controversies, edited by Kates. Particularly relevant essays are probably the Introduction, Furet's piece, Baker's piece, Tackett's piece, Markoff's piece, and Hunt's piece. The second edition has two pieces (I think) on postcolonialism which might help.

For the Sonderweg, there is really nothing better than Eley and Blackbourn's The Peculiarities of German History. The introduction perfectly sets you up for the argument.

For counterfactuals, Ferguson's Virtual History is good. I might also suggest What Might Have Been by Andrew Roberts and Our Shadowed Present by Jonathan Clark for fairly self-avowed conservative historians.

For quantitative history, I would suggest "The Political Unconscious of social and cultural history, or, confessions of a former quantitative historian" in Sewell's Logics of History.