IO540, my apologies, the headwind effect is indeed second order, as you said all along.
Specifically, about 60hx degrees where 'h' and 'x' are the head and cross winds expressed as fractions of the TAS. So for 225/30 wind the error is about 2.5 degrees, which is comparable to the accuracy of flying a heading manually.
Eliminating the error would involve something like this:
1) Take off, then set the DI by visual reference to something outside. This does not have to be very accurate. Plus or minus 10 degrees should be fine.
2) Fly a 'one minute square', ie one minute E, then S, then W, then N, using the DI for direction. Analysing the GPS tracks afterwards will give you headwind, cross wind, and TAS.
3) Fly one leg, North by the compass, long enough to get a good True Track estimate from the GPS track afterwards.
4) Solve the wind triangle problem to get the True Heading, then apply the best estimate of the local magnetic variation (Whopity's point, I think) to calculate the actual magnetic heading.
A lot more complicated, and probably pointless, but more accurate with respect to the wind!