In this case, however, it's easier to use weaken.

Simply change

sub new {
    my ( $class, $href ) = @_;
    my $self = { dbi => $$href{dbi} };
    $self->{helper} = Helper->new(
        helper_function => sub {
            $self->this_is_from_A(@_)
        },
    );
    bless $self, $class;
}
[download]

use Scalar::Util qw( weaken );

sub new {
    my ( $class, $href ) = @_;
    my $self = { dbi => $$href{dbi} };

    {
        weaken( my $self = $self );
        $self->{helper} = Helper->new(
            helper_function => sub {
                $self->this_is_from_A(@_)
            },
        );
    }

    bless $self, $class;
}
[download]

(By using the same name for the weakened variable as the unweakened one, we avoid using the wrong one by accident.)