You could look at hylafax+/iaxmodem/t38modem and the various mail2fax scripts that that culture will expose to you, as far as I can tell they are all “prior art” and just use open source technology and effectively bypass FreePBX/Asterisk.
t38modem uses asterisk’s t38 “pass through” ability but requires a T38 compliant VSP.
iaxmodem will do T30 faxes over a good g711 connection.
both need spandsp compiled and working with asterisk compiled against it (most Distros do that if you use a Distro).
When it’s all working you can setup your mailserver to pass(pipe) appropriate mime content through a script that will use “sendfax” to send your emails to a fax machine.
If you want an http(s) GUI client, then add Avantfax to the mix.
They all work fine for me and have for years.
If you have windoze impaired clients add
and they can just “print” to your hylafax server.