I think, if you scale it up a bit and adjust the weight for increased friction losses, you may be able to do it considering available materials, but the required knowledge would be anachronistic. I mean, they could also have built compound bows and repeating, lever operated crossbows, but weren't smart enough.
For the string you could use a chain, or just a thicker rope. A presume a bow string has similar requirements and was available tech. For the frame, spindel/shaft and string, I don't see a problem as they appear to be scalable, not restricted by dimensions/weight much. Wood, natural fibers and steel would do. In theory, medieval people in "The East" had readily access (as in no advanced processing required etc.) to everything needed to make highly elastic rubber (for the original supersonic trebuchet), but it would be an anachronistic material AFAIK. You may conceive a variant using maybe an array of bows for quick release energy storage.
Wood is plenty strong (in relation to weight), but the arm may critically depend on modern materials, as both supersonic trebuchets use specifically advanced materials there (structured titanium and carbon fiber composites IIRC). For a heavier arm, you would probably need a prolonged acceleration phase, which would be doable (for Stanton's trebuchet), but I am not sure the forces wouldn't tear lower strength:weight materials apart, or if a more voluminous arm may be able to overcome the drag. I could see the fat compressed part of the arm being aerodynamically shaped wood (like a wooden propeller; although "aerodynamic" anything may be considered anachronistic), and the thin tensioned part made from a thin band of steel/bow string.
I think, another potential show stopper may be bearings. Not sure you could have the rotational speeds of the arm without precision (ball) bearings and modern lubricants. 2342 RPM would probably the fastest spinning thing at the time. (I wanna see a washing machine gone trebuchet-y, now.)
But really, it would be just a curiosity in medieval times too. What's the point of supersonic speeds in a device you can hardly aim precisely? You could deliver the same or more energy by throwing a heavier projectile at slower speeds, without excessive aerodynamic losses.