I installed carpet for a while and did boat interiors for correct craft for a while also. I know of no way to eliminate the flex of the carpet. My thought is to widen the flex over a wider area to keep from putting so much stress on the line of the hinge. What about a piece of electric taper laid on the hinge evenly and glue over the whole thing. We always used Weldwood contact cement applied wet and liberally. Never had any issues with it coming loose.
Creativity is just intelligence fooling around