The solution was to use Foiply and set up two different accounts, one for the VERP and one for the actual From address. Also, I needed to setup a Gmail account to do the forwarding, since it uses a constant VERP, while VERP from Google Groups changes with each message. Steps:
Create a team account on foiply.com. You need to support two incoming addresses, ...
Google Docs and Sheets add-ons1 or Google Apps Script could help solve this issue.
Regarding add-ons, they could interact with the user Gmail mailbox, i.e., there is an add-on, Email Forwarder2, that allows to create rules to forward emails even of old messages and include options like "include headers" that could be helpful in this case.
Add-ons are built ...
I believe Ring Central Fax will do what you want.
I actually called them up to answer this question and here's the answers I got:
While their site talks about attaching things to email and faxing them, they will fax the body of an email message (apparently as a / in lieu of a coversheet)
The account has an address whitelist
... but the email addresses on ...