I'll weigh in with my two cents, since I actually took the reverse path you're contemplating: I sold my D7100, and bought a D800E. I was very satisfied with the D7100, and after using the D800E for some time, in my opinion both are excellent birding cameras, and you can't go wrong with either one. Here are my very particular reasons for switching; these may not all be relevant to you, and many of these honestly turned on the fact that I also have an Olympus OM-D E-M1 and a Micro Four Thirds lens set for my "small" camera kit:
-Having gotten used to the large EVF of the E-M1, I wanted the larger viewfinder of the D800E, which has turned out to be useful for planning and framing BIF shots
-The D7100 has very good high ISO performance, but at a pixel-peeping 100% view, it has a lot of luminance noise, which in my opinion was too prominent to give a large resolution jump over the D800E, even when comparing 24MP vs. 15.4MP
-24MP DX was testing the resolving limits of many of my lenses, particularly on the telephoto end. Even my 500mm f/4 needed to be stopped down one stop to get maximum sharpness, and just about all lenses I used lost their luster when a teleconverter was attached. Thus, I figured I had nothing to lose in going back to 15.4MP of DX resolution, as I was previously quite happy with the D7000's 16MP images.
-I wanted better fast wide-angle prime options, in particular the 28mm f/1.8G, which is one of my most-used non-birding lenses. For the D7100, the Sigma 18-35mm f/1.8 is actually as good, or better than most FX primes optically, but it's pretty large and heavy for a non-telephoto lens, and its focusing can be finicky.
-My DX kit didn't have much of a performance jump (aside from AF) over my Micro Four Thirds kit. Since I'd be using the same telephoto lenses, the wider angle portion of my FX lens set wouldn't be much larger or heavier than what was available for DX, and with a boost in image quality and low noise performance, a better compliment to my Micro 4/3 kit with its tiny lenses.
-I probably have a mild to moderate case of Gear Acquisition Syndrome. I wasn't able to pass up a good deal on a D800E, which led to my selling the D7100 and all my DX lenses to make the switch.
-The D800E in 1.2x crop mode creates an image about the same size (~25MP vs. 24MP) as the D7100, at a slightly lower frame rate (5fps vs. 6fps), with better pixel-level noise performance. The D7100 only has a 1.25x crop advantage compared to a D800E in 1.2x mode.
I'm very satisfied with the D800E. For bird photography, the D7100 probably still has a small edge in resolution, thanks to its higher pixel density, and the fact that although the D800's viewfinder is larger, distant birds are still more magnified in a D7100 finder. I find autofocus performance to be essentially the same between the D800E and the D7100. But otherwise, I prefer the D800E for all non-bird photography; I find the shallower depth of field and high resolution very pleasing. That said, if I had to pick one sensor size and system, the most cost-effective answer I'd be reasonably satisfied with would probably be a kit built around a D7100.
Here are some shots I've taken with the same lenses on both cameras. They are uploaded at full size, so you can pixel-peep to your heart's content.
D7100 & 80-400mm AF-S:
https://flic.kr/p/gjyvKH
https://flic.kr/p/fgVxsZ
D800E & 80-400mm AF-S:
https://flic.kr/p/mZLt29
https://flic.kr/p/nfRf8Y
D7100 & 500mm f/4 AF-S II:
https://flic.kr/p/mQpH1N
https://flic.kr/p/iSQwBJ
D800E & 500mm f/4 AF-S II:
https://flic.kr/p/ksVmVB
https://flic.kr/p/nPLZyT
The good news is that I don't think you can go wrong with either choice. If funds allow, and you can tolerate the increased weight and bulk, consider getting a larger telephoto lens. The new 80-400 is a very good lens, but a used 500mm f/4 AF-S, or the Sigma 500mm f/4.5 is definitely a step up in reach and sharpness without becoming unmanageable.