In addition to some perfectly reasonable ideas already presented, here's the way I do a similar thing. In my case, I'm going to use $baz rather than $self->{baz} to show that it's a lexical. Or, more to the point, that $baz may not be known until you're three levels deep inside _internal_sub and completely dynamic.

sub alpha; # unchanged from your example.
sub beta {
  my $self = shift;
  my $coderef = sub { $_ eq $gamma };
  $self->_internal_sub($coderef);
}

sub _internal_sub {
  my $self = shift;
  my $coderef = shift;
  foreach ...
    if ...
      if ...
        if ...
          local $_ = $baz;
          push @foo, [ @bar ] if $coderef->();
        }
      }
    } else {
      local $_ = $baz;
      push @foo, [ @bar ] if $coderef->();
    }
  }
  local $_ = $baz;
  push @foo, [ @bar ] if $coderef->();
  \@foo;
}
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I prefer the $coderef->() version over &$coderef because the former says "dereference this code reference, and call it with no parameters" while the latter says "call this piece of code, (ignoring prototypes - not relevant with code refs), passing in @_." The former is what I'm doing, not the latter.

However, since you have three places you call this coderef, it may be better to just pass in the parameter:

sub beta {
  my $self = shift;
  my $coderef = sub { $_[0] eq $gamma };
  $self->_internal_sub($coderef);
}

# and ...
  push @foo, [ @bar ] if $coderef->($baz);
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Hope that helps,