I had not played much since reporting this bug as it was severely impacting any fun I might have had during a match. I have only played one match just now, I think the performance problem is gone. I hope others and mine RPD files have helped narrow it down, big thank you to you /u/RiotTony and the other Riot employees.
The RPD files and discussion here have been invaluable. I'll probably write up a Tech Blog on the cause of this once we've done a full Root Cause Analysis (where we figure out what went wrong, why, and how we can either avoid it in the future or better respond when it does happen).