You do get involved in some interesting projects.
I was giving this some thought
and I was wondering under what circumstances an arrow could outrun a bowstring. My initial thought was that it couldn't because you're always applying a force to the arrow so you should always be accelerating the arrow. However on reflection I decided that didn't have to be true. As the bow and bowstring accelerates, viscous drag (air resistance), and possibly other resistance forces, will increase so that if the string is moving fast enough the bow may not be able to accelerate the string at all - and if the bows force is sufficiently non linear the string might actually slow down allowing the arrow to outrun the string.
Having said that it seems that doing the experiments in a vacuum might give a misleading result.
You clearly need a pressure, temperature and humidity controlled test chamber.
It also seems that bows where the draw weight decreases as you pull back the string ought to have the advantage of a much smoother acceleration of the arrow - and possibly increased accuracy as a result.
Now, back to the original question. I think that trying to get some real data is the right approach and I think your strobe approach should work. I did wonder whether you could approach the strobe the other way around and add flashing LEDs to the bow - or even in lines behind the bow but I suspect it might be difficult to see the bowstring. I think that approach is used in some other sports to track players actions.