I have an older Prince machine w/ a similar clamp and occasionally string wooden or badminton racquets w/o open throats. When that happens, I just put the C-clamp on the inside of the frame pointing outward instead of the normal way you have it which is on the outside pointing inward. You might think it would get in the way of the last cross but I've never had a problem.
As Irvin asked and I'll ask too....Does that C-clamp have a P on it?
Back to using POGs. Why did I ever leave you dear friend?