In regards to Library of Congress, it boils down to "do we believe them to be competent enough to recognize their own limitations?" If yes, then we should hand them over. There are a number of scenarios afterwards:
1) They execute lesser akuma as soon as they get out of sight of us, if they don't think they can hold them
2) They vivisect lesser akuma for knowledge
3) They trade lesser akuma to their chinese / japanese equivalent agencies who have specialized facilities
4) They will hold lesser akuma in captivity
5) They will refuse taking lesser akuma
In regards to holding the akuma in captivity - we know that they can be contained in a magic circle that Harry can produce. I am fairly sure that the Library has talents working for them. Whether they have full-on wizards is a question, but I would bet they do. As such, I think it's fairly plausible that they'll be able to hold lesser akuma. Iron-cast magic circle, plus mostly automated (akuma don't produce techbane) supermax system guarded by talented people and / or non-human personnel, if they have any, should be able to handle them.