Unfortunately, there is not currently a way to schedule this process during the evenings. We had talked about doing this, but never did.
There are two things potentially you could do to help with this. One is that you could put the high and low watermarks closer together, that way it migrates data in smaller chunks and may not necessarily overwhelm the server as much.
The second thing is that there's a TierMigrationSpeed configuration option in the ShredHostService.exe.config file. This option was put in place to attempt to set a rate in MB/s that data could be transferred from one tier of storage to another. We attempt to schedule the tier migrations into the future so as not to overwhelm the server based on this option. If you were to set this option to a lower value, it should slow down how often the migrations are scheduled, and potentially leave more resources for processing incoming studies.
The final suggestion would be to look at your hardware configuration, and potentially having dual NICs where incoming studies come in on one NIC and your filesystems are on the other NIC to isolate the traffic. Tier migration is obviously a heavy network activity, and this would help.
Steve