AFAIK you can just “dial” IP address on the ATA, e.g. 12*50*3*4# to initiate a call to the 126.96.36.199. This is probably the simplest solution.
>Also: is this doable with private IP addresses?
No, you will not be able to use it if both ATAs are behind the NAT. You will need some sort of proxy in between them or special NAT/firewall which will allow voice traffic through.