Ok, I guess this falls into the category of "you learn something new everyday."
I've never seen replacement screens offered for the 5D III. Even companies known for selling screens for many cameras (e.g. KatzEye) don't sell screens for the 5D III. I had chalked this up to the active focus points.
But when you posted that you were able to find sites that sell screens for the 5D III (I know people for whom the lack of replacement screens was a major complaint on the design of the III), I did a search and was able to find two different sources. One site outlined the installation steps... none of which involved electronics (ok, so they're clearly not making the active display work by embedding it in the screen -- which was my understanding for "why" the 5D III couldn't have replaceable screens.)
I did a check on the 1D X (because it has what's largely the same focus system as the 5D III) and I noticed that the 1D X _does_ list optional focus replacement screens for that camera. So much for the active LCD theory. The focus points are apparently not really on the screen... they only appear to be on the screen. (BTW, this now makes me wonder why Canon doesn't offer replaceable screens for a 5D III... considering the rather STEEP boost in price over the cost of a 5D II which _does_ offer replacement screens.)
One seller (on Amazon.com) claims to offer the 'same' focus screen as the original part (so it's not advertised as an optional screen... it's just a replacement screen.) What's odd is that the photo of the screen does NOT match the photos of the screen I see at another site where they illustrate step by step replacement instructions (and the camera in their photos is clearly a 5D III). I'm not sure if they erroneously posted the wrong product photo (it happens) OR if they're not actually selling the correct part.
As I have never before heard of anyone replacing their screen -- you would be among the first (you'd be *the* first person I'd ever heard of doing this). I'd be cautious. You might phone Canon service to inquire as to the cost to have your damaged screen replaced by them. For other cameras, Canon's screens run less than $50 and the replacement is pretty simple. The 3rd party screens were about $100 (except for the part I see listed at Amazon (which is rather yellowed looking and doesn't have tabs in the correct locations)... which was about $25.)
Ouch, how did you scratch it? Are you sure it's a scratch or might it be something easily cleaned off? Can you see the "scratch" when looking through the viewfinder? If it's sharp, it's on the focus screen. If it's blurry, it's on the mirror. You should never touch either with anything... they are very fragile and easily damaged (leave it to a professional if they ever need cleaning).
Any possibility it's just a hair? You might try carefully "puffing" it away with a bulb blower.
If the screen has to be replaced, I'd send the camera to Canon (or take it to one of their authorized repairers) for replacement. This would probably be cheaper than one of those third party screens and certainly safer than trying to repair it yourself. Note: another consideration, you will void the camera's warranty replacing the screen yourself.
There are third party screens available, even for 5DIII and 7D that have the active matrix, transmissive LCD focus screens. In most cases, the third party screens are installed by removing a few screws, but still isn't a job for someone unfamiliar with working on cameras. The third party screens usually offer enhancements for manual focus and/or special alignment grids, sometimes a little brighter viewfinder. However, there's a cost. If you add a split image manual focus assist rangefinder dot to the center of the screen, your spot metering will no longer be accurate. Typically the third party screens are 3X to 5X the cost of the Canon screen, so it would probably be cheaper just to have it properly repaired.
Third party focus screens for most models can be seen at www.focusingscreen.com. Also search for Haoda and Brightscreen. There are some others... but I'm drawing a blank right now.
In all likelihood, a minor scratch on the focus screen (or a little dust or fuzz) will have no effect on auto focus, metering and certainly can't have any effect on your images. When you take a shot, the mirror flips up and covers the focus screen during exposure. So anything on the focus screen or mirror is merely a nuisance, most of the time. You could just learn to live with it. I guarantee you there will be dust and other stuff on there eventually, too.
I've replaced the focusing screen in the 6d. I do not recommend the third party screens with the added focusing aids. You lose the use of the center area where the focusing aid is for metering. If you use a lens with a smaller aperture than the f2.8 such as a f4 or f5.6, you lose the use of the center area as the screen darkens in low light situations. Auto focusing works in all situations. It is also very difficult to align the Canon center circle overlay with the center focusing aid. I use and recommend the Eg-S super precision matte screen available most everywhere for less than $40. The center area is free, there is no issue with the Canon overlay and even with a lens like a f4 or f5.6 the screen is still usable in all but a few low light situations. The only wrinkle with the 5d Mark 3 is you have to remove two screws to get to the focusing screen. Still, not a big deal. The Eg-S gives you a much better idea of what is in focus especially when manually focusing. I still use auto focus and the red light confirmation for manual focus lens.