SpriteBatch will render at whatever size you tell it to: just use one of the overloads that takes a destination rectangle parameter, and you can scale that destination to whatever size you like.
A common technique for fill-intensive shaders like blur processes is to shrink the source image to half or quarter size, blur that, then scale the blurred result back up to the original resolution. This can save a lot of pixel shader work with minimal quality loss, since the output image is supposed to be blurred in any case.
You should be aware, though, that laptop graphics cards often don't have much fillrate, and fullscreen blur processing needs a lot of fillrate! Have you tried running our bloom sample (from this site) on this card? How does that perform?
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