Gmail – How to set up Gmail to change the from/to address when automatically forwarding email

I have two Gmail accounts: my normal Gmail account (me@gmail.com), and an account for my school that is hosted through Gmail (me@school.edu).

I get lots of emails on both accounts, and some people send me emails in multiple contexts (friends that I work with). I have all emails from me@school.edu automatically forward to me@gmail.com, so that I have a single point of searching and archiving and don't always have to swap back and forth between accounts to do some basic things. In me@gmail.com, I try to filter all of these forwarded emails in my School label and then archive them from my inbox.

This runs into a problem when the email that goes to me@school.edu does not include my direct email, like for mailing lists or some forms of bcc. Gmail forwards the email with the original from (sender@school.edu) and to (list@school.edu or otherlist@groups.aol.com), and so it ends up unread in my me@gmail.com inbox.

How can I change either the to: address on the forwards to me@gmail.com OR the from: address to me@school.edu, so that I can set up a filter that works in all situations?