My apologies, but I seriously think you're making a mistake.
A lot of people on this site throw out recommendations based on personal intuition of what they think will happen without backing it up with data from tests/benchmarks/etc.
I can explain the reason for this, not to sound like an arrogant jerk, but some people just have certain technical knowledge, and to a degree do not require benchmarks to know.
But I can show you some benches that will give you an idea. Now this isn't going to be "perfect", but I think it reasonable enough for this example.
For all intents and purposes an i3-2300 is essentially half an i5-2400 as far as cores go. The clock speed of an i3-2100 is 3.1GHZ and so is an i5-2400. The i3 is a dual core, the i5 is a quad. In this test they used a 7970 which at the time was the most powerful video card on the market, safe to say the GTX 680 is much more powerful however.
http://www.tomshardware.com/reviews/gaming-fx-pentium-apu-benchmark,3120-3.html
Pay particular attention to those two benches. Clearly the lacking 2 cores is limiting the FPS both average and minimum. Now you move on to Battlefield 3, but the benches there are only single player, and SP BF3 really doesn't care what CPU it is. I'm in a hotel room with my laptop, so I don't have my desktop with me, I had a link bookmarked about BF3 multiplayer, but I'll ask you to take my word for it, it indeed has been benched and BF3 multiplayer in ultra settings demands a quad core CPU.
Moving on to just cause 2, again you see both the minimum and average are affected.
http://www.tomshardware.com/reviews/gaming-fx-pentium-apu-benchmark,3120-5.html
Starcraft II... 2 cores is holding the FPS back
http://www.tomshardware.com/reviews/gaming-fx-pentium-apu-benchmark,3120-6.html
Dirt 3 doesn't really care either way
Metro 2033 I've benched myself, the performance is all over the place, that has some really loopy coding, Russian made game.. what do you expect?
Now all of these differences aren't very substantial no, but if maximum FPS is your concern, again those benches were done with a weaker card than a gtx 680, the holdback on the FPS will be more profound.
What do you mean by saying there is a limit to how important fps is...
When you watch a movie in the theater, or on a DVD you're watching it at 24FPS. Now video games are a little bit different, but its reasonably fair to say that the human eye does run into a limit where he/she is no longer capable of noticing a difference. For example, the average person is not going to notice a difference between 80 and 100 FPS (assuming you had a monitor that could display that)
If my avg fps is 60 then that means I'm going to have dips of much lower, maybe 40, 30, or even 20. But If my average is closer to 100 then my minimum will probably be higher as well.
Thats not necessarily true. Go back to metro2033, even with powerful CPUs, its all over the place. It depends on how the game in question is coded.
Only if the cpu is too wimpy to drive the beastly 680 gpu, right? Okay, well I have all these people saying "get an i5 theres no way an i3 is fast enough", but no evidence to back it up. I'm not trying to be an ass or anything I just would like to see something that supports this claim.
I think I covered it pretty well in this post.
Edit: Another thing you have to consider is background programs and the affect they have on gaming. A quad core is going to be less likely to be bogged down by these programs while gaming (antivirus, open browsers, etc)