Hi Lynn:
Your school has that policy for a reason. I don't know what it is but is among the majority of colleges that have adopted similar policies, probably mostly attributed to avoiding accusations of unfair classroom practices.
Let's make the assumption that learning is the purpose. If the class is averaging low, then what about some remediation, tutoring, or review, then repeat the test (or one similar). In this instance, no extra credit is given, just extra instruction. Voila! Class average improves, and students have earned every darned point on their own.
Regards, Barry