You could try storing the flag outside of the object itself:

package whatever;

my %customDestroy;

sub new {
    my( $class, $flag, ... ) = @_;
    my $self = {};
    ...
    $customDestroy{ $self } = 1 if $flag;
    return bless $self, $slass;
}

...

sub DESTROY {
    my $self = shift;
    if( $customDestroy{ $self } ) {
        ## special behaviour
    }
    else {
        ## normal behaviour
    }
}
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I think package lexicals should persist long enough to outlive instances, though I haven't tested the hypothesis.