but that still make a circular reference because of $self

You could stick close to your original solution with a method call if you pass a weakened copy of $self to the subroutine reference:

use strict;
use warnings;
use Devel::Cycle;

package A;
use Scalar::Util qw/weaken/;
sub new {
    my ( $class, $href ) = @_;
    my $self = { dbi => $$href{dbi} };
    my $weak_self = $self;
    weaken $weak_self;
    $self->{helper} =
        Helper->new( helper_function => sub { $weak_self->this_is_from
+_A(@_) } );
    bless $self, $class;
}
# ....the rest remains unchanged.
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That way, this_is_from_A has full access to the $self object (which isn't needed in your example so far).