Application requires IP address, need it to go to domain

A reverse proxy is a good idea, but may be limited by protocol.

I'd solve this problem by setting up a VPN (which provides security - if the app is that badly written it does not do a DNS lookup it's probably not using decent crypto either). Using a VPN means your addresses can all be rfc1918 ( ie private ones ), and the app won't even care when the external addresses change.