Monks, Is it possible to retrieve the attribute name that is currently being set in a builder method - similar to what caller() does?

No, it is not possible to get that information. However I suggest you use the approach described below as it will not only reduce code duplication, but also make it that much easier to add similar attributes later on and in subclasses.

package This::That;
use Moose;

has 'FOO'  => (is => 'rw', lazy => 1, builder=> '_build_foo');
has 'BAR'  => (is => 'rw', lazy => 1, builder=> '_build_bar');

with qw(SQLConnection);

sub _build_foo { (shift)->_build_generic('FOO') }
sub _build_bar { (shift)->_build_generic('BAR') }

sub _build_generic {
  my ($self, $attribute_name) = @_;
  my $sth = $self->dbh->prepare('select $attribute_name from table whe
+re x=? and y=?');
  $sth->execute(1,2);
  return $sth->fetchrow_array;
}
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Note too that this approach also makes it possible to have a custom FOO or BAR builder in a subclass. With your previous version you would have needed to change the builder method pointer and do other hackery that would lead to issues later on.