The exact (within a degree or so) runway headings are published, but I would need to extrapolate them to north-south and east-west. I don't really see how I would do that on the ground. The only way I could do this that comes to mind is to use an aerial protograph, and mark n-s and e-w on the photo. As I see it, I'd have to then transfer this to the ground at the airport. That would mean wandering around on the ramp, either painting lines or making suitable marks.
If I were trying to build them a compass rose, this might be an approach, but it sounds like a lot of work just to calibrate one compass. I have my doubts about the city accepting this, or understanding what I was doing..
But I do think you have the start of a viable approach here. If it were my private airport, this is probably exactly the approach I'd use to build a compass rose. I may think on this a little.
Unfortunately, a hiking compass is of unknown accuracy, and thus unsuitable to calibrate an aircraft compass (my A&P side is showing, I know). I do have surveying equipment, left from another project, but I think this is another significant undertaking. I'd need some reference points, etc etc. I think the photo approach would yield almost as accurate results, with less hassle.