Wrap the Background setting in Dynamic. The variables in the Background option seem to need to be global.  Either like this:

    background = 1;
    Manipulate[{}, {m, 1, 10}, 
     Grid[{{Dynamic[background = m; 
         Plot[Sin[x], {x, -m, m}, ImageSize -> 200, Frame -> True, 
          ImagePadding -> 30]]}}], Paneled -> False, ContentSize -> {0}, 
     Frame -> False, Alignment -> Center, 
     AppearanceElements -> "ManipulateMenu", 
     BaseStyle -> {Background :> Dynamic@Hue[(background - 1)/9]}]

or perhaps like this:

    Manipulate[{}, {m, 1, 
      10}, {{background, background = RGBColor[.80, .80, .90]}, 
      ColorSlider},
     Grid[{{Dynamic[
         Plot[Sin[x], {x, -m, m}, ImageSize -> 200, Frame -> True, 
          ImagePadding -> 30]]}}], Paneled -> False, ContentSize -> {0}, 
     Frame -> False, Alignment -> Center, 
     AppearanceElements -> "ManipulateMenu", LocalizeVariables -> False, 
     BaseStyle -> {Background :> Dynamic@background}]