Byregot's Blessing does indeed remove Inner Quiet, so you cannot use it together with Rumination.
Personally, I will use Byregot's over Rumination whenever possible.
Rumination is for those times when I miscalculated and don't have 24 CP for Byregot's at the end. In those cases the 40 ~ 60 CP from Rumination gives me the chance to perform a few more actions to try to save the situation, but it usually doesn't equal the quality gain I could have gotten from Byregot's Blessing. ^^;
