Or, since constructors can have any name, what if you defined sub testing_constructor?   This method would only be invoked by the module-testing code.

To my way of thinking, the tests of the base-class (having thus been made possible by this alternate, special-purpose constructor) would of course be limited to functionality that was strictly in the base ... and that could, meaningfully, be “tested” in such abstract isolation.   (Resist the temptation to have “tests to have tests.”   Tests must actually make sense.)

You might consider coding the abstract base-class so that it had such things as ... a new method that blew up, and stub-methods that also blew up.   Just so that the abstract class wouldn’t do anything harmful if accidentally invoked in a senior moment brief episode of amnesia...