You can get that string through that grommet. No matter how blocked it looks, you can do it. Carefully insert an awl into the hole from the inside of the frame and push the blocking string out of the way. Let the awl just sit in there for a minute or two. Place the sharpened tip of the string right up against the awl where it enters the grommet hole. As you slowly pull the awl out, push the string in at the same time. If you can get the slightest string tip to show through on the other side, you can grab it with your needle nosed pliers. If you can't get a hold on it, try pushing it through with your pliers. Grasp the string very close to frame and push. If you can move it just a few mm., you can grab it from the other side.
Sometimes when you are stringing, you will be making great time and something like this will come up and take several minutes to resolve.
NOTE: If you are using an awl with some delicate string, like gut, be very carefull not to damage the string when pushing the awl into the blocked hole.
I don't think leaving out one cross would damage the frame, but then again I'm not an engineer. In the "old days" people would do patch jobs. If you have already finished and tied off, you might be able to string that last cross with a single piece of string. I guess the problem would be finding two grommets to tie off on.