Ip – Are private IP addresses unique

It doesn't answer your question, but may answer your need: I fear the only option would be to have all peers register their local IP addresses on a server directly connected to the Internet (without NAT). Each peer could then query the server for the local and translated IP addresses of others. If their translated public IP addresses are in the same range (allocated in the same AS), they could then try a direct connection through their local IP addresses.

Of course, with end-to-end IPv6 connectivity, there would be no issue...

The two kinds of NAT your question is referring to can be categorized as destination NAT, and source NAT.

Destination NAT will typically change a connection to your router from the ISP direction to a destination target that's inside your network. It's also commonly referred to as port forwarding. This lets you expose a service that would normally be inaccessible inside your private network.

Source NAT will change a connection through your router from your local network to a source target that is the router itself, allowing multiple hosts to use a single public address.

NAT can perform other translations, but few others are useful except in niche cases. Despite the phrasing in your question, a source NAT that you describe is not helpful. Using a global address that is not functional on the router will have no good effect. No other device or protocol would keep track of that, so all replies will be forwarded to the device that should have that IP, which would know nothing about your setup and promptly drop them. If you want to use a different IP, get it set up and functional on the router first.