Mmm... it's not independent of supply voltage, in fact it is proportional... I suppose it could be constructed in a way that minimizes that, however. Example: perhaps the yellow wires entering the movement are to *de*magnetize a permanent magnet, thus tending to reduce the movement's gain when the input signal is made larger by the elevated supply. This would only work over a modest range, because deflection goes as the product of magnetic fields. If the movement (rotor) field is proportional to Vsupply, by making the stator field proportional to (1-k*Vsupply), the product becomes Vsupply - k*Vsupply^2, where k can be adjusted to null the fluctuations around nominal supply.
Just a guess. There might be other, better ways of doing it.
The "resistance bulb" I would guess is something with a well-defined tempco, like nickel or constantan. A PTC or RTD type thermistor: fairly weak (resistance not quite proportional to Tabs), but reliable. A thermocouple, of course, would require cold junction compensation, which would be more complicated.
The nice thing about a Wheatstone bridge is, its output (i.e., the voltage across the "span" of the bridge) is zero when the circuit is balanced. If the meter movement is not just a movement, but a feedback mechanism which adjusts one resistor until the voltage reads zero, it would indeed be independent of supply voltage again; a variation in supply would modulate how strongly the mechanism is able to correct, and therefore control the linearity and error of the reading. I do see what looks like an adjustable rheostat on top, but it looks like the movement is enclosed, so there would be no way to adjust a bridge resistor; and there's no indication on the schematic that there is such a mechanism.
Tim
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