Auto Renew Membership
Welcome to CiviCRM SE.

I think you have quite a complicated membership processing structure there (very frequent payments), but the basic requirement seems to be process membership update structure from an external file (I presume you get the paycheck info as a CSV file) is quite common. If you don't need to record the payments at all I think you can do a membership import to update the membership expiry dates (you would need to do a bit of processing on the file first to indicate the appropriate expiry date.

If you want to record the payments (as I do) then its a bit more difficult and I haven't yet found a good solution (How to bulk renew membership is similar).