I made an account to comment on this particular issue. I made my own investigations, since it was really frustrating to not be able to play this game in good conditions.
1: You have noticed this already, but the AMD cards are perfectly capable in the game, the problem comes from specific locations or scenes where the GPU usage drops very low.
The issue has already been discussed here, but I wanted to add what I found.
Crysis 3 low FPS drops & low GPU utilization with RX 480 This thread includes several videos of the issue being reproduced on AMD GPUs, and not being there on much less powerfull Nvidia GPUs
2: I have managed to replicate this issue in another game, Kingdom Come Deliverance. This is particularly interesting as the game also runs on CryEngine (a more recent version). And in fact it is possible to solve it. Look at this video Kingdom Come Deliverance - r_BatchType perfomance - YouTube
The issues are very similar, good FPS until you look at a certain scene, then GPU usage drops to a low level.
Sadly I have very little knowledge about what this setting exactly does, but it does fix the performance, I also tried it myself.
I have no idea if it fixable for Crysis 3, I fear that it is not. I have tried everything, but I have not managed to find a similar console command on Crysis 3, perhaps this version of the engine didn't support it yet.
Since I am new on this forum, let me know if I posted on the right place.
Hopefully someones reads this.