Problem is that an i3-2100 doesn't use the same socket as your current CPU (i3-540) so you would have to upgrade the motherboard too. But if you really want to go this path might as well get an i5-2500K or i5-3570K instead.
Also you didn't say which version of the 650 you have. As ish416 said the standard 650 (non-Ti) is not that fast, maybe you are expecting too much out of your video card. At what resolution are playing BF3 5 min? Maybe it's normal to get 15 FPS with a 650 if your resolution is high and you use a lot of AA and AF.
i know i have to change my mobo too ASRock H61M-VS will be fine and cheap, so you do think the 2100 isnt the best idea? cuz its the cheaper. i have 650 non-ti
and i play bf3 on a 19" monitor so the resolution isnt very high, and as i said it doesnt matter cuz i get same fps on any gfx settings or resolution. when i open the game i get 50 fps on ultra and high res(also tried low gfx and low res)... then i play for a couple of mins and it starts lagging to 15 fps.
and yes gfx card drivers are updatey
my friend lend me his 9500gt since i cant use the 650.. and i get about 20 on ultra so believe me the 650 cant have 15 on bf3.