You are definitely going to be using the 2gb card close to the limit if you are playing BF3 max @ 5760, but it shouldn't need more.
http://www.bit-tech.net/hardware/201...-hd-7850-2gb/4
2560x1600 needs more than 1gb, so 5760x1080 definitely needs more than 1.5gb, but you can see the 2gb cards holding their own at 5760.
The 1.25gb 560 Ti 448 stays steady at 2560, so that is probably just the right amount of VRAM, scaling up that implies you need around 1.9gb for 5760x1080.
You can see that the 2gb 7850 retains 77% of it's performance from 2560 to 5760, while the 3gb 7950 retains a similar 71%.