Wow that’s absolutely super cute @mattbrailsford, and a very sweet technical and mechanical achievement!
I would say try to place a 100uF or larger low voltage 6.3 or 10V (to keep the size small) on the power leads of the affected servo and if possible the servo that is the cause. Right on your servo jumper board would be good.
It seems like the servo is interpreting a change in the pulse width of the neutral PWM output. You could correct this in software by applying a bit of counter rotation pulse width. So for example if it always seems to spin a bit backwards when activating another servo, it might be too close to the backwards throttle point so you can trim it by moving the pulse width a bit closer to the forwards throttle point. This neutral (no motion) zone is called the deadband zone, in which there is a range of values that don’t cause movement. The wider this zone is, the less sensitive the servo or motor control system is to noise and voltage change.
If you have an o’scope, you can monitor the pulse width of the affected servo while operating other servos and see if it’s just noise, a voltage drop or if the Core is changing the pulse width a bit incorrectly. If the latter is true, I believe adding caps won’t help you.