To commend on your performance, it would significantly increase if you had two sticks of RAM for dual-channel mode. See these reviews:
http://www.guru3d.com/articles_pages/amd_a10_6800k_review_apu,11.html
http://www.tomshardware.com/reviews/a10-6700-a10-6800k-richland-review,3528-6.html
All of the games they mentioned and tested (ie Tomb Raider, Bioshock Infinite, Metro 2033: Last Light) are much more intense on your system than Battlefield 3, and they are are newer games too. The Guru3D system manages just short of medium graphics detail at 1080p, without any overclock, and thats with small levels of AA on. AA being the biggest performance drain you can pretty much use.
Tomshardware review takes a different approach and does low settings, but with FA and FXAA on, again AA is a huge drain on performance.
Took a little work to find this:
http://www.xbitlabs.com/articles/cpu/display/amd-a10-6800k_6.html#sect0
Without overclocking, not using enthusiast RAM really, they managed 26 FPS on BF3 at 1080p low settings. After an overclock, you will probably be running over 30FPS without much effort. That still probably isn't going to be enough to satisfy you into tossing extra cash into your system, so I would say instead try to get several advancements for your system to get better results.
Like above, I'm going to assume a max budget of $135
http://www.amazon.com/dp/B006WAGG14/?tag=pcpapi-20
http://www.newegg.com/Product/Product.aspx?Item=N82E16835214054&nm_mc=AFC-C8Junction&cm_mmc=AFC-C8Junction-_-na-_-na-_-na&cm_sp=&AID=10446076&PID=3938566&SID=
http://www.newegg.com/Product/Product.aspx?Item=N82E16814150688&nm_mc=AFC-C8Junction&cm_mmc=AFC-C8Junction-_-na-_-na-_-na&cm_sp=&AID=10446076&PID=3938566&SID=
These three items are:
Ram 4GB $36
CPU Cooler $10 (After MIR)
R7 260x $87.99 (After discount code and MIR)
Total: $133.99
If you do that, you would have a much better system as you could push your CPU up to 4.6Ghz easy (I have one running at 4.6Ghz without any voltage increase on a similar but even worse cooler). You will greatly improve RAM performance and overall system feel from the RAM. In gaming you will be able to play most games at 1080p on high settings, sometimes without AA. It also comes with 2 free games from AMD.