It isn't clear if the copy processes you are running are part of the general server workload or something that can be scheduled for when there is little load on the machine.
If copying is done frequently and you are using a kernel > 2.6.25 you may find ionice of use.
From the man page:
This program sets or gets the io scheduling class and priority for a
program. If no arguments or just -p is given, ionice will query the
current io scheduling class and priority for that process.
For example, to copy a file using the 'idle' scheduling class, you can do something like:
ionice -c 3 cp /tmp/example.gpg /tmp/another_example.gpg
-c 3 switch equates to selecting the
idle scheduling class, about which the manual states:
A program running with idle io priority will only get disk time when no other
program has asked for disk io for a defined grace period. The impact of idle
io processes on normal system activity should be zero.
This seems to me to be precisely what you are looking for.