No you will have no problems with that CPU. Heck clock-clock it's faster than any Bulldozer by a longshot and BD ca CF/SLI heavy hitters no problems. There may be a few games that will take a small hit but an overclock will take care of that no problem.
May I ask you what games are you having trouble maxing at 1080p? i would think a 670 would demolish 1080p gaming aside from mayb FC3. i hace an SOC gtx 570 still and rarely encounter a game I can't max
Getting 4x Msaa playable in 1080p can be a challenge for a single 670. (hitman, farcry, arkham city off the top of my head)
I added a 2nd one recently. Can get 70fps+ on FC3 at 4xmsaa (8x gets mad laggy and blurry even though fraps gives me 45 fps-- think its driver side issue idn) Hitman Absolution i get 52+ at 8xmsaa
I think the brunt of the benefit of a 2nd 670 would be seen at 1600p levels though.
Also be aware that adding the 2nd gpu will bump your temps up by ~10C give or take, just something to remember.
That's crazy. I just tested those games you mentioned and 4x MSAA is completely playable at 1080p in Arkam and Hitman with my SOC 570. THe 670 is a good amount faster. I never tried FC3 but I know I can't max that game anyway. 8x MSAA is completely unplayable on the 570 but 4x does pretty well. THe 670 has 800mb more Vram than my 570......