The D800 will give you 15mp in DX mode (4800 x 3200 pixels), compared to 12mp from your D300. IMO, the D800 in DX mode, has the same resolution as a D7000 but with better ISO quality.
Your 300mm with the TC20E III would be a very nice 600mm lens. Getting more reach than this will cost a lot more money. Getting close to small birds is difficult and for me is more luck than skill. I often find a place with a lot of bird activity and then stand still and wait for them to come to me rather than trying to pursue the birds. If you don't move much you'll be surprised how well this works.
The D4 is 16mp in FX mode, so you would need a really long lens to be able to fill the frame with a small bird. It would not give you as many pixels to crop into as the D800. Probably less than your D300.
Here's a good example of the high ISO capabilities of the D800. This was shot in FX mode, ISO 3200.