Change background color of MP4

Hi - I have an MP4 (no AE project to work from) that has a logo graphic animating over a gradient background that goes from a light grey (hex D0CFD1) to a lighter grey (hex FDFBFE).  The logo graphic is a combination of black and some very saturated colors. 

I've been asked to make the background white.  When I try the "change to color" effect, I get weird striations across the background.  If I try to tweak the tolerance, it then starts to include the graphic.  If I try using a curve, it effects the graphic too much.  

Is there a way to change this background to a solid white without changing the logo animation?