How Frankfurt's example refutes compatibilism

1. (Premise) Frankfurt's example shows that if compatibilism is true, then the Principle of Alternate Possibilities is false.
2. (Premise) The Principle of Alternate Possibilities is true.
3. Therefore, compatibilism is false.

Of course, Frankfurt's example was meant to show that whether or not compatibilism is true, the Principle of Alternate Possibilities is false. But because, as has been shown for instance by Widerker, the example assumed the existence of a sign that determines one to engage in a particular action, the argument failed. (There are variants on Frankfurt's example that have been proposed that fix the problem, but whether they succeed is controversial.) However it is easy to see that Frankfurt's example has refuted compatibilist versions of the Principle of Alternate Possibilities (e.g., Hume's version on which if x freely does A then had x wanted not to do A, x would not have done A).
It is worth noting that (2) was accepted by classic compatibilists like Hume and Ayer (with a compatibilist reading of "able to do otherwise"). Of course, they might well have abandoned the Principle of Alternate Possibilities given Frankfurt's example. But perhaps they shouldn't have. For the Principle is deeply plausible. Instead, they should have abandoned compatibilism.
A potential response is to modify the Principle in a flicker-of-freedom way. This is open to compatibilists as well as incompatibilists.