you very frequently have prior information which you want to bring to the problem, but of course you don't want to bring in any prior information which is not neutral on the controversial issue that your study is supposed to shed light on

Well, let's be very explicit about that then. A good report will:

• state the relevant priors used for everything that the study is not directly about but which are still relevant,
• remain quiet about priors for what the study is directly about, giving likelihood ratios instead.

More mathematically, suppose that you make certain assumptions A which, in full completeness, are not just things like "I assume that a certain sample has been dated correctly." but "I put the following probability distribution on the dates of this sample." This is very lengthy, which is the inconvenient part; although if you make simplified assumptions for purposes of your calculations, then you would put simplified assumptions in your text too. So it shouldn't really be any more inconvenient than whatever goes into your analysis.

But what you are testing is not A, but some hypothesis H (that the ancestors of Homo and Pan split after they split from Gorilla, for example; notice that this only makes sense if A includes that these three genera are really clades and that evolution of these animals really is a branching tree, although these are pretty common assumptions). And you have some evidence E.

Then in addition to A (which goes into your introduction, or maybe your appendix; anyway, it's logically prior to examining E), you also report the likelihood ratio P(E|A&H)/P(E|A&!H), which goes into your conclusion. Then maybe you also state P(H|A) and calculate P(H|A&E), just in case people want to read about that, but that is not really your result.