Definintely do not try to Odin back to 4.3 if you installed any 4.4.x bootloader and modem.
Before doing anything else let's be absolutely certain of the bootloader and modem your phone is running on. (I was not certain what you used based on the link you provided. There are a number of debrick images, bootloaders, and stock ROM dumps on that page.) Since you have TWRP installed you can use the terminal emulator built into it. Run these commands for a definitive answer.
Code:
getprop | grep version.baseband
Was the error message you received from attempting to install CM11, a 4.3 ROM, or some other ROM? Specifiy the ROM that gave the message, please. I would expect that message from trying to install a ROM that was built before the release of your installed bootloader.
BTW: AT&T did not release any complete 4.4 ROMs compatible with Odin. Most of the Stock 4.4.x ROMs in the forum are from XDA folks who captured the images from OTA prior to flashing and then built them as Recovery flashable zips. But, for the i747M there may be Odin flashable 4.4.x series Stock ROM. That could be the root of the issue.
First off, huge thank you for assistance with this old device.
I looked for the terminal emulator and I thought I clearly remembered that it was built into TWRP but it seems it doesn't exist. I can't get to any terminal. I can however tell you that this was the bootloader file that fixed the hard brick once flashed: d2att_I747UCUFNE4_bootloader_modem_4.4.2_7-29-14.zip
After that, per the OP instructions, I went ahead and also flashed this modem: I747UCUFNE4_ModemBaseband.zip
Given that I was previously on CM11/4.4.x, I attempted to flash I747UCUFNE4_Stock_Rooted_Odex.zip with no luck, then the latest CM11 nightly for the S3. I'm not planning on going back to 4.3 given all the warnings everywhere.
It might be useful to know that what I was previously on CM11 and was trying flash Lollipop/CM12 which is when I first ran into issues. I gave it 20min to boot and figured it wasn't going to boot so I went back to load the nandroid but it seems I had cleared one step too many and had no nandroid. So I simply googled for a stock Odin flashable 4.4 ROM which is when I hard bricked it...likely because it was an i747M ROM as you specified and I came to the same conclusion. This is my back up phone and I was going to use it but I wanted Lollipop since my newer phone has it. When I last updated the S3 to CM12 successfully I told myself I'd never flash it again because it works 100% and it's just a back up. But I couldn't resist
At this point, it loads up into TWRP and Download mode fine.
This is the error when attempting to flash CM11 or I747UCUFNE4_Stock_Rooted_Odex.zip: E: Error executing updater binary in zip...
I haven't tried other ROM's but I consider CM ROM's pretty standard and I've never had an issue with them so unlikely another custom ROM would work over CM.
UPDATE:
Figured out how to get to the terminal...
getprop ro.bootloader gives me: I747UCUFNE4
getprop | grep version.baseband yields: no response from the phone. Just shows what I typed...
UPDATE 2:
So I believe this may be a similar 'you can flash up but can't flash back' kinda things. Given that the bootloader and modem I currently have on there is compatible with the Lollipop ROM's, I attempted to flash the latest CM12 from here:
http://forum.xda-developers.com/galaxy-s3-att/development/unofficial-cyanogenmod-12-t2968854
It successfully flashed so I went ahead and flashed the appropriate Gapps and recommended SU update, wiping cache and delvik between zip files. All went through fine. I've been waiting about 5min now for it to boot...all I see is the CM man with the evil eyes and "Samsung Galaxy SIII" above him.
UPDATE3:
Waited about 12min, nothing happened, held power till turned off, ejected microSD, booted into TWRP, wiped cache and delvik again, factory reset, reboot and WOW! It worked! "Android is updating... installing app XX of 105"
PS:
The link to the GAPPS is broken on the more up to date 5.0.2 page by matrixzone. I used the GAPPS file found on an older CM12 thread here:
http://forum.xda-developers.com/galaxy-s3-att/orig-development/rom-cm12-d2att-alpha-builds-t2959561
Seems to have worked.