Bandwidth is different than rendered frames. However, the card still has to push those frames out to my 120hz monitor, which means double the bandwidth. It's absolutely a factor in my setup.
120hz tvs are a gimmick. They're completely different than a true 120hz monitor. TVs will only ever accept a 60hz signal, the second rendered frame is done, as you mentioned, on the onboard hardware. With a 120hz monitor, however, a special Dual-link DVI cable is needed for the necessary video bandwidth as HDMI cannot support it.
If you're using an HDMI cable, your maximum bandwidth is either 340MhZ if you're using an HDMI 1.3 cable, or 165MhZ if you're using a 1.0 spec cable. Know what that means? Your displayed image could be starving itself. This can translate to blurred images or loss of image quality. Since you're at 368mhz, if you're using a good 1.3 spec HDMI cable, it's probably negligible or unnoticeable. To say it's not a factor though, would be incorrect. If you're using an older spec HDMI cable, you're affecting your setup without a doubt.
Also, at 60hz your monitor is only displaying 60 frames per second, so I would also argue you're starving your eyes and wasting the other rendered frames, as your monitor is dropping them

Our eyes can see well over 200fps, and there's not really a proven limit, as our eyes don't really see in "frames." There was a study a while back done by the airforce with pilots done in a dark room, and they would shorten the time frame for displaying a light to see if the pilots would be able to see it. They got down to at least 1/200 th of a second and the pilots had no troubles seeing it. That translates to their eyes being able to pick out one frame with the test running at 200fps, more or less.
Just to be clear - the card renders what it renders. However, bandwidth DOES come into play. a 120hz TV =/= 120hz monitor. Twice the bandwidth IS being pushed to the monitor, and it WILL display 120 frames per second, rendered at the card level and NOT the monitor level.
For me, that's why I couldn't run at a higher res with my current setup. My goal was to play BF3 at 120fps @ 120hz, and thankfully my setup got me there. Plus I don't have 1k to drop on a set of 2 680s
I really dork out on the AV stuff, as I used to do it for a living a while back.
So again, like I said earlier, for me 680s would be underwhelming. For my setup, I actually have a higher score than you do. Throw in another monitor, however, and I'd upset the "balance of the force" and then I'd have to drop another 1k for those puppies. Not a road I want to go down right now. Especially if I want to finish my AR-15 anytime soon
Oh, and if you didn't figure it out already, I like to debate. Don't ever mistake me for a one-upper though, that's not my style. I just like to talk with my LAW buddies and learn new things.
You never did mention if you only had one card before? What monitors do you have? I'd like to see what fps you're pulling in BF3 at that res. Are you using MSI Afterburner or anything that has an On Screen Display where you can see that? I'm sure it's up there, as BF3 isn't really THAT taxing.
Also, are you running BF3 on two monitors then? How does the bezel between both monitors affect your view when playing? I've always wondered this, and is probably the main reason I haven't pulled the trigger on a second monitor yet.