Currently, the only way to SIM unlock the Sprint Galaxy Note 3 for use on GSM networks outside of the US is for Sprint to unlock it via OTA. And for them to that, the phone must be in an active Sprint account.
Unlike the Sprint Galaxy S4, there's a method to SIM unlock the Sprint S4 without Sprint having to do it for you. Unfortunately, this method has been tried and tested on the Sprint Note 3 and it didn't work.
Putting a different ROM on the Sprint Note 3 won't help...and totally unnecessary for use in other countries. You're at the mercy of Sprint to get it SIM unlocked.