Rather than drill into the plate, (which might release some stresses, and as you say, produce distortion), how about making a frame to carry it, and put the handles on that?
With regard to recovering the damage, how about this as a suggestion?
Take a smaller plate and by whatever means, get it as flat as possible, and then blue and scrape it using an undamaged area of the surface plate. When you are happy with the result, use this "sub master" to check the areas that you have been carefully scraping to remove the burrs. You will have to live with the damage below the general surface, but don't add to it by going wild with the scraper around the dings!
Howard