Very cool! You can see from your nice diagram that the sender is very similar to an IVR. Ignoring the OP function, it’s set up so that the heater loads the input voltage (coming from ignition voltage passed through the gauge) and in doing so disconnects itself via bending of the bimetallic strip as it gets hot. If ignition voltage is high, heating is faster and duty cycle is reduced, and vice versa. So the effective time-averaged voltage supplied to the circuit is regulated. This works only because the bimetallic instruments are so slow.
The OP function is an add-on, in effect. The distance between the contact points decreases with increasing OP, so that the heating coil is operating more of the time–a greater fraction of the duty cycle at a given ignition voltage. So both ignition voltage and OP change the duty cycle.
I tested this on the bench, similar to what you did except with a regulated power supply to provide the voltage, and an ammeter so I could observe when the contact points closed and opened. I tried about 10 Smith’s OP senders, of which about half worked. With a working one, I could set the voltage to, say, 13 volts and vary the input pressure (from a N2 tank) to test its operation. Conversely, I could set a pressure, and then vary the voltage to test how well it regulated against change in voltage. They worked perfectly, giving the same (correct) pressure reading over the range of 9 to 14 volts IIRC (this was years ago).
I suppose these were made for use on cars without IVRs–otherwise a resistive unit would suffice, like the temp or fuel sensors. Anyway, Jaguar eventually started using gauges that didn’t require voltage regulation, and these odd critters became obsolete. Although a PITA, I think they are very classy and very Jaguar.