You don't want white, if anything you want black. If you use white then the reclection/shadow or your light fixture is going to show up on the marquee. Typically you want to just leave the wood unfinished as it's a happy medium in terms of color and reflection.
The light should fill it just fine, just be sure to mount the light at the deep end and have it pointing towards the shallow end.
But honestly, why won't you just plug in a florescent tube, stick it in the box, and see how it does?