So here's the skinny. I SIM-unlocked my UNACTIVATED T-Mobile Note 3 via Chainfire's RegionLock Away app.
I bought the Note 3 SM-N900T direct from T-Mobile with the sole purpose of using on AT&T - without thinking about it being SIM-locked to T-Mobile. Of course I found this out when I put the AT&T SIM in out of the box - only to be reminded that it was locked and I needed the network unlock code from T-Mobile.
I also didn't want to activate a SIM on T-Mobile (spending at least $10 for a week-long plan) just to turn around and spend $25 on an unlock code. So after seeing Chainfire talk about fixing the Region Lock on the Note 3 I decided to give the following a test:
1. Put the T-Mobile SIM (unactivated, unregistered) that came in the box into the Note 3
2. Rooted via Chainfire's CF AutoRoot
3. Bought RegionLock Away app
4. Ran the app, and instead of rebooting - I said Reboot Later (I THINK THIS IS VERY IMPORTANT)
5. I then powered off the Note 3.
6. Took out the T-Mobile SIM and replaced it with my AT&T SIM
7. Turned on the device - BOOM! no Region Lock, and NO SIM Lock. I was able to setup my APN without any "wrong network" errors, and it's working like a charm.
I highly recommend you purchase Chainfire's app and give it a go. Follow the steps EXACTLY as I laid them out above. I can't guarantee anything, and YMMV - but this worked for me.