There are two main methods of keeping the water out of your nose while doing the inverted dolphin kick. The first is to exhale constantly through the nose and you can see this in action in lots of videos of top swimmers. The second is to block the nostrils, which some do by rolling the upper lip back to cover them, and the other way to do this, and a simpler one in my opinion, is to wear a nose clip. You may find it uncomfortable at first but you will get used to it. I never wear a nose clip in races but always in practice, both for face down and face up swimming. You will see a lot of elite backstrokers wearing nose clips.
I no longer suffer from sinusitis and sneezing after swimming since I decided to use a nose clip.