The SS controller is sitting at the other side of the room from the London controller using the same equipment hence there is no requirement for the London controller to verify the Mode C again.
Makes no difference where the SS controller is, or what equipment they are using - transfer from one radar unit to another means that the next controller can assume (unless told otherwise by coordination) that the Mode C is validated and verified.
As such there is no need to report passing level again. The controller retains the option of asking you to report passing level at any time - this may be due to garbling etc.