But Jas, BF3 is highly GPU dependent game and it just need a CPU which can handle the GPU without getting bottlenecked. There are certain CPU dependent games too (Like GTA 4) which stress CPU a lot. Also OP has mentioned that he wants a system for longterm use.Also it offers better overclocking potential too because it is a black edition CPU and comes with a better Heatsink.
So combining all these cases I guess Phenom II X4 955 is a better choice here for long term use.
But I also agree with you that HD 6770 is also good choice.
My suggestion will be Phenom II X4 955 + Sapphire HD 6750.
AFAIK, HD 6750 from Sapphire is available in sub 6K price is ouple of online shops. here is the link for HIS HD 6750 1 GB GDDR5 @ 5.8K: HIS - HD6750 Fan Graphic Card GDDR5 ( 1GB ) ― Lynx - The Audio , Electronics And Computer Online Shopping Store In India.