Easy, reward the improvement in test results, not the test results. This could also encourage good teachers to go to lower performing schools (since they might see the largest gain (and hence pay rise) there).
for one, do we have a hidden reserve of these "excellent teachers", and if so, can we convince them to leave their more lucrative jobs elsewhere?
Yes, pay them more. If you are very skilled at mathematics or engineering, you might be a good maths teacher. However you can get much better money working in Google, than working as a teacher. So pay maths teachers Google level salaries. You would have to pay teachers different amounts depending on what they teach.