I think if the customer is making a lot of changes, the 'automatic' solution you are looking for is probably over-engineered.
However, I have been a bit blinkered by sticking solely to MySQL.
In principle, it should be possible to generate the modulus of the week number (previously used to filter the toptenlistID) in PHP.
Then use this to select a row number from the recordset.
Personally, I have no practical application for this, but I hate to be defeated.
If in the meantime Jason comes up with something, then I rest easy!