Do you have two HDDs [i.e. platter-driven hard-drives]?
One safe way to copy massive amounts or massive-sized files in by using TeraCopy or anything similar that will generate on-the-fly feedback as the copying progresses. Another, do the copying in "clumps", do not pick all the files at once. After a few "clumps", give the process a short rest, several minutes may be enough, then continue.
I'm not sure if Harddisk Sentinal or anything similar will track the temps of external HDDs - have to experiment and find out.