I think you should make high-resolution photos of all the pieces and donate those to the Chrono Compendium first, for historical purposes! (Unless we already have a visual archive of those?)
Sometimes a collection can fetch a higher value than the same pieces sold separately. Sometimes it's the other way around. It largely depends on whether the completeness of the collection will deliver significant value of its own in the minds of potential buyers.
I would try selling the collection first. If that fails, break it up.