The dye has probably penetrated the finish. If you can't rub it out with a light abrasive (much finer than the sandpaper you tried), I think you probably need to apply more white. You most likely do not need to strip off the old finish (though there is a chance the die will bleed through the new finish.) The existing finish is probably pigmented lacquer if it's the factory finish. You can simply apply new white pigmented lacquer.
That said, there is no "universal white." I have about a dozen different white lacquers in my kit and often have to use two of three of them to kick in a little yellow, brown, or red. And you also need to get one of the right sheen. Further, a lot of white furniture comes with a very transparent glaze on it. Since you don't want to try this yourself, contact a local furniture repair / refinishing company, who can most likely do this on-site with aerosol cans.