I believe what you are referring to is what’s called a silk thread suspension. One end of the thread attaches to the post and the other end passes through a hole on the post and extends down to and wraps around an arbor. The pendulum rate is adjusted by rotating this arbor which either shortens or lengthens the effective length of the pendulum by winding or unwinding the thread. The silk thread suspension was found on most early french movements. If the movement has a Brocot escapement, ie; one that has the pallets and escape wheel on the outside of the dial, then it should have a clutch type beat adjustment. You could just push the pendulum to one extreme or the other to set the beat. If not a Brocot then my guess is that the crutch may need to be bent in the correct direction. Your cautious approach is a wise one as forcing the pendulum can easily bend the delicate teeth. There is often another way to adjust the beat in these clocks. These round movements are mounted in the case by slipping into a drum or ring and are secured in place by tightening the screws that hold the back door onto the clock. If the adjustment needed is a minor one then loosen these screws, start the pendulum swinging and slowly turn the front bezel while listening for a steady beat. When you think you have it let it run for a minute or so until the pendulum settles down to a normal swing. If it sounds good then tighten the screws and your all set. These movements are very high quality and generally show little wear after many years of use. The plates are thick and the pivots, pallets etc. are hard and highly polished. Beautiful movements! If you decide to repair it then pay particular attention the the small pivots when assembling. As I just mentioned, they are hardened and can break if not lined up in the holes when tightening down the plates. As long as you don’t force anything you should be fine. They actually assemble quite easily and are a pleasure to work on.
Hope this helps Ed,