It isn't actually the measurement itself. It's when the metrics end up tied to rewards and penalties that people start to game them. [1]
What you could do is measure things and then, when unit 15 is above average and unit 9 is below average, figure out why and let everybody know what works and what doesn't.
Which also has the side benefit of improving your metrics. Because if you see that unit 15 has the best numbers and you treat this as an undifferentiated "unit 15 is better and we don't know why but let's reward them" metric, you can miss things like, unit 15's district has a higher population density and they're actually below average after you take that into account.
Investigating the sources of success and failure without assigning personal consequences to them allows people to be honest about why they succeeded or failed. And then if anybody has actually found the secret to success you can share it with everyone else.
[1] Although you do have to be careful not to make "collecting metrics" a thing that eats half of each employee's work time.
That's key. You never get them right the first time, so improving your tracker is even more important than improving the tracked.