Staying with Nikon, the Nikkor AF-S 17-55mm f/2.8G DX IF-ED is faster but would put you in the same place for reach. You could instead switch to an FX body, but no single f2.8 lens seems to cover your needs & it would be a very expensive switch.
If it were me, working to your budget, I'd would skip the f2.8 idea & stay with DX glass. I think that f2.8 is semi-pro/pro territory with a price to match & with a DX body, you are not going to get the full benefit of such a big expense. My suggestions in post #3 I think best suits your requirements & budget.