What about stopping the copy job or moving the log files waiting to be restored to a separate temp folder on the same drive ? Not ideal but at least the job won't be able to restore any more files - its going into suspect mode because you're stopping a log restore before its complete.

Are you using ms log shipping or your own method ? We use powershell for our log shipping and it works really well - Worth checking this link;

http://www.sqlservercentral.com/articles/powershell/99832/

Cheers

Vultar