Put your internal apps on real IP addresses with real DNS and just firewall them?
This just seems like adding duck tape to a self-inflicted problem.
(Not that having chrome use 8.8.8.8 is a good idea either. As is often the case with hacks, either on its own works fine but the two interact poorly.)
That being said, it very well maybe time to find a better alternative :).
As for DNS, it has a very elegant hierarchical delegation system, such that organizations can control their own while also ensuring that all the world's DNS servers give consistent answers (and if you need short names for your internal systems on your own network, that's what search paths are for).