I have always considered using a radar gun. I have one from my baseball-playing days.
There would need to be 2 considerations...well, 3, actually. The 1st would be just simply acquiring one...because at around $79.99 or so which was what my radar gun cost at the time they are not all that less expensive than a chronograph. 2, one would need a second person to actually attempt to gun the arrow’s flight (and they would need to be well-versed enough with using a radar gun to capture an accurate reading). And 3rd, the recorded results would then need to be converted from miles per hour to feet per second.
All in all, I would think that there are other fairly accurate alternatives to a chronograph...and they all might actually provide reasonably accurate results. And from what I have seen with a lot of people’s chronograph speed declarations, FPS numbers gathered by other means would be scrutinized equally as much as those from a supposed chronograph!!