The black borders will also darken it - if the blocks are amenable to sashing, that will also darken it. So will the black thread you plan to use for quilting it.
When you get a chance to post some pictures, that will be helpful - because then we can also see how 'white' it actually is.
Sometimes we 'see' what we are looking for - if something has exactly the same ratio of colors in it - sometimes we 'see' more of the light/bright color than of the rest of it. Example: Dark purple and bright yellow. The yellow will be a lot more 'dominant' than the dark purple, even if the surface area is the same.