Is there a way to get the behavior I was looking for, without either changing method names or the class hierarchy?

No.

The method call, Qprime::bar has been resolved already. Now another method call, Q::SUPER::bar, needs to be resolved.

You are using the wrong tool. If you got what you say you want you would have a loop repeatedly calling Q::bar.

Usually SUPER is used to add functionality to a method:

sub Child::foo {
    SUPER::foo;  # do parent functionality
    blah;        # add child functionality
}
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Changing the order of the @Qprime::ISA elements might be a possible solution.

If the hierarchy is carved in stone, i.e. you can't even change an @ISA, then you might make explicit calls: $self->P::bar.