Private-use networks connected to the public internet are an ugly hack. If you don't have a public address, you don't have a public address in any meaningful sense, so any effort to figure out "your" address is doomed from the start.

Particularly in these days of IPv6, just give everything a notionally publicly routable address and then every device can know its real address (of course this doesn't have to mean you actually route public traffic to every address if you don't want to).