As you might have guessed from my fumbling, I'm nowhere near to an expert on this topic. It does seem like this is something that should be possible. If you like, you could submit a bug report/feature request to the WiFi101 library's issue tracker:
I did a little searching and didn't find any existing issues on this topic. Interestingly, there was a pull request that seems to have been intended to add the ability to get the current DNS IP address:
You can see here a function that returns the DNS IP:
Unfortunately, that pull request was using files from the Adafruit_WINC1500 library rather than the WiFi101 library and so it had to be rejected. The Arduino developer did say this about it though:
We'd be happy to review any equivalent changes that are made to WiFi101.h and WiFi.cpp
So it sounds like, at least at that time, Arduino was open to adding that functionality.
There are actually two ideas here. The first would be that once you have dnsIP(), you can do this:
WiFi.config(staticIP, WiFi.dnsIP(), WiFi.gatewayIP(), WiFi.subnetMask());
But the library has these function overloads of WiFi.config() that make us think we can leave off parameters if we don't want to change them from the existing value. The other idea would be to add that functionality to the library.