There’s only been a few dungeons where we have bowed our head like this before.
The most memorable is after defeating Edda in Tam-Tara HM. Because of what happened to her, and because we knew her from very early on in our journey. Our character also repeats this motion after defeating her on Floor 50 of PotD, and when we defeat PotD’s Floor 100 boss. I think the latter is us just contemplating his words—where he asks us if we would not desire for us and ours to cast off the fetters of mortality and escape death.
As mentioned, I think this is our character’s way of mourning the fates of the Sin Eaters. Knowing what we know about how they were once real people. I believe that’s also why a lot Sin Eaters are usually named with “Forgiven X”—what they have become is no fault of their own, so we find it in us to forgive them for the sins they committed as a Sin Eater.
However, this is just my interpretation of it.