Since the volume is constrained, the question should be "is nitrogen somehow exempt from Gay-Lussac's Law?". That's the law that describes the relationship between pressure and temperature. But your point is correct. Nitrogen is used where heat becomes a factor in tires because it deviates less from the ideal gas law. Water has strong intermolecular interactions that cause it to deviate from ideality. Nitrogen's pressure curve vs temperature is closer to linear while water's pressure curve is exponential.

Since bicycles don't see the heat load that tires like those used in race cars do (passenger cars don't really see those kinds of heat loads either), using dry nitrogen would have little effect. To bring the discussion back to spoon brakes, putting that kind of heat load on a tire would make nitrogen beneficial. Putting that kind of heat load on a bicycle tire would be stupid but nitrogen would help a little in that situation.