Peekay,
I have to say that I don't get it. pV = (m/M) R T gives you the pressure at a given temperature, not the change in pressure. If you had two identical flasks, and filled one with water vapor and one with dry air, the pressure in the first would be higher. No surprise there -- water is less dense, and therefore a given mass has to be compressed more to fit a given volume.
But (m/M) in your equation doesn't change with temperature. R, of course, is also constant, and the volume of a tire (V) we can also assume is constant. So if you double the absolute temperature (T) in your equation, pressure (p) has to double as well -- none of the other numbers change. And that's true whatever the value of (m/M) is.
It is true that neither water vapor nor air is an ideal gas, and they probably do expand at somewhat different rates. But it isn't as simple as a ratio of molecular weights, and I'm virtually positive the difference is much less profound.