As much as I hate to break this to you. Yes, your CPU is the bottleneck. Even a i7-3770k isn't fully capable of utilizing dual GTX 770s properly until around the 4.4-4.5Ghz mark. SLI eats IPC for breakfast. And that is something the FX-8350 does not have. You won't notice good scaling on dual high end GPUs with AMD unfortunately. Single card, that would be different. But dual cards, yeah.
And BF3 multi-player eats CPUs for lunch. It is a highly CPU bound situation, that really does show whether your processor is capable of it or not. Even super high end (3930k+) struggle with it without a fair overclock. The FX-8350 will flounder in BF3, while inherently causing a CPU bottleneck in the process with SLI. Combine those two, and you are in for a world of hurt.
Considering your cooler, I'd pick up an i5-3570k/3770k/4670k/4770k minimum.
Just to give you an idea what I'm talking about, read this article: http://www.anandtech.com/show/6985/choosing-a-gaming-cpu-at-1440p-adding-in-haswell-/7
I specifically linked that part of the article because Civ 5 is a fully multi-threaded game that will eat up as many as you can produce. It still does have GPU scaling the higher up you go in card counts. And for this review's sake, the HD 7970 would be in place of your GTX 770s.
That part of the review shows that even though the entire game is HIGHLY multi-threaded, the FX-8350 is still inherently a bottleneck. You are in that same situation with BF3 & multiplayer BF3. You just can't fully utilize your GPUs with that CPU.
If you do decide to switch, you can probably pick up a board like the z77x-UD3H (pretty much the cheapest/best Z77 board for the performance) pretty cheap and go SLI on it. I'd stay away from the ASRock z77/z87 Pro3/Pro4/Extreme3/Extreme4.