Firstly, It is ESSENTIAL that the clock case is firmly fixed to a wall and the case must be set absolutely vertical, both front to back and side to side.
Most clock instructions say this but it is usually disregarded on the grounds that "this is only a test"!
The Synchronome was designed to allow the pendulum to swing as freely as possible and the gravity lever if properly set, provides only just sufficient energy to keep it swinging. If the case is not fixed then the heavy pendulum will cause tiny movements of the case and these use energy required to keep the pendulum swinging.
Secondly, the setting of the gathering arm with its jewel is vital. It might seem logical for it to be set so that it fully engages the countwheel while it pulls that wheel forward but it was not designed that way and causes far too much load on the pendulum, which as I have said, was intended to swing as freely as possible.
The gathering arm should be set so that it only engages a tooth of the wheel just barely sufficiently to pull it forward one click. This is mentioned in all the instruction I have seen, but it is not sufficiently emphasised.
www.awco.org/clocks/ Look for the Synchronome section and within that the adjustment notes. This has a copy of the notes by A Mitchell which are widely recommended.