I'm guessing Twilio is one of the cheapest and you're already set up with them, but if you'd consider RingCentral at about 1.7c/page which is reasonable then you can still keep Twilio for other things and just use RingCentral's email-to-fax plan.
So you'd add an email address on your contact records something like
firstname.lastname@example.org, and then use it the same way you'd use any other email address, attaching the pdf or whatever.
There are others besides ringcentral, but that's one I'm familiar with.
UPDATE: One workaround for tracking successful transmissions using the above is to use a dedicated sending address like 'email@example.com', and then set up the email processor for inbound email processing on this address (see https://docs.civicrm.org/sysadmin/en/latest/setup/civimail/inbound/#autofiling-email-activities-via-emailprocessor). Then when ringcentral (or whatever) sends the transmission receipt it will get automatically filed into CiviCRM. It won't get filed on the specific contact without some extra programming, but it would at least get filed in CiviCRM.