Short answer: there is no advantage to filling the tires with nitrogen; the thermal expansion of nitrogen is not significantly different from that of air.
Longer answer: the ideal gas law, pV=nRT, is indeed just an approximation, but to tailor it to the behaviour of real gases, you don't just tweak the value of R, you use the more accurate van de Waals equation:
(p+n2a/V2)(V-nb) = nRT
where the parameters a and b reflect the deviations of a given real gas from ideal behaviour. If a and b are both zero, the gas will behave as an ideal gas.
For nitrogen, a = 1.408 and b = 0.039; for oxygen, a = 1.378 and b = 0.032
For comparison, for CO2 a = 3.6 and b = 0.04, while for helium, a = 0.034 and b = 0.023.
The conclusion we draw from this is that, while neither nitrogen nor oxygen is quite ideal, their deviations from ideality are almost identical; thus the expansion coefficient of pure nitrogen will be almost identical to that of air.