You circuit is novel and should conceptually do what you want.
However the devil is in the details and, in trying to implement it with real parts, I had a problem with generating the required stable gain and offsets with the normal variation in battery voltage and the high common mode voltage of the signal.
One of the problems is that the bridge circuit output varies with the applied voltage when it is not balanced, as is the case here.
So I generated a, perhaps odd looking circuit (LTspice simulation below), that stabilizes the voltages with a programmable shunt voltage reference (U2) and tried to minimize errors by running the sensor current through an op amp feedback resistor to directly measure the current.
Due to feedback, Op amp U1a biases the temperature sensor (simulated by current source I_Temp) at 4V from the 4V Ref2 voltage.
The op amp feedback also forces the temp sensor current to go through feedback resistor R1 to generate the output voltage proportional to the current (temperature).
The current through R2 from the 6V Ref voltage, offsets the 255uA bias current at 0°F to give a zero output at that point.
To avoid interaction of the two signals, I separated the function of the the oil light from the rest of the circuit by op amp U1b acting as a comparator which drives a transistor to turn on the oil light when the oil switches closes.
The simulation below starts with the oil switch (red trace) closing at the 0.1s point and opening at the 0.5s point.
The temperature current then starts increasing (yellow trace) from 255µA and going to 388µA, with the output voltage (green trace) starting from near 0V and going to ~2.4V at the maximum temperature.
Due to component tolerances, some tweaking of R2 (offset) and R4 (gain) may be needed to get the proper output offset and range.