Can you explain the 3pm on Tuesdays issue? My sister works for LLS and she said their servers get very slow at a precise time every Tuesday. Not saying it's the same bug, but what was the solution in your specific case?
So in that case, I guess either run the job with a lower priority and see if that helps, or execute the job more often so it doesn’t have to catch-up all at once one time per week, or rewrite it so that it performs I/O with smaller chunks of data at a time and sleeps for a little while in-between reading or writing chunks of data. Basically, do something so that you no longer have this one huge job consuming all of the IO bandwidth for several minutes every week.
There was one periodic job that we moved from the production server to work off the daily backups instead of the live server.
It's not something anyone can diagnose from what you say, it could be anything, even weirdness such as a hardware fault kicked off by something else (office cleaner plugging something in?) causing power spike RF interference affecting the network causing mass packet drops and retries (ok, unlikely but it's not impossible, I've heard of such).