fortunatly for you, i have tested both those cards 550 and 6770, with a i5 2500k and a 8Gb DDR3 system.... unfortonatly neither one of those cards pulled a satisfactory result, they both stick to very good fps in lower settings (70+fps easely with medium/low shadows off) but stutters too much to be playable.
Anyway, the 550 is very much superior to the 6770.
For OVERRAL use, i can highly recommend the 550 for NOW. this card can run BF3 on high/ultra settings with no problems, and il2 1946 runs easily.
6770 will settle for similar performance to the 550, with graphics on medium in BF3, and will have no problem with 1946.
FOR COD use, i wouldnt recommend anything below the 560, and a 2GB card seeems to be mandatory. SO with that budget... i dotn think yo ucan pull it.
Pheraps a 2gb version of the 6850 is the best you can get, with will probably be worst on OVERRAL than the 550, but ti will be better for COD due to the 2 gb memory