I'm the original poster of this problem from a year ago. All the suggestions about trapping what is happening at the time are good. Since you know when it happens, scheduling sp_who2 and sending the output to a file might tell you what you need to know.
Also check that maintenance plan job you have running at midnight to make sure it has no additional steps other than the integrity check. Look for anything that runs Transact SQL or a stored procedure. And - try to find out if the application performs any sort of maintenance at midnight.
You could change your full backup to run at 12:05 (right after the event, whatever it is) until you figure it out.