1. Why do all of the BNCs have no GND connection? Each should have the individual cables to the ground. Each signal line and the corresponding ground line should be twisted together.
2. This looks the (usual) oscillation of the V-I conversion stage but I can't tell anything as I don't have the circuit diagram of the whole circuit.
3. In a certain case, putting some capacitance at the feedback may help. Read P.11 of the data sheet of LT1125. Try to put some capacitors from 20pF to some larger one whether it changes the situation or not. I suppose the bandwidth of your sensor can be ~1kHz. So putting a capacitance less than 10nF still has no effect to the servo.