Corion is right that the method should be called on $dbh (though in my test it worked on $sth too, so I'm not sure that's the problem here). Note the documentation: "The last_insert_id method was added in DBI 1.38." - what version do you have?

use warnings;
use strict;
use feature 'say';
use DBI;

my $dbh = DBI->connect(
    "DBI:mysql:database=testing;host=127.0.0.1", $ENV{USER},
    'barfoo', { RaiseError => 1, AutoCommit => 1 });
$dbh->do('DROP TABLE IF EXISTS coacttran');
$dbh->do(<<'ENDSQL');
CREATE TABLE coacttran (
    id INT PRIMARY KEY AUTO_INCREMENT,
    foo VARCHAR(256) 
);
ENDSQL

for my $i (1..10) {
    my $req = qq{ INSERT INTO `coacttran` SET foo=? };
    my $sth = $dbh->prepare($req);
    my $row = $sth->execute("bar$i");
    say $dbh->last_insert_id;
}
[download]

I spun up the test database with Docker the same way I showed here.

Also note that I don't know how you're generating $_pstmt, but you really should use placeholders! See Bobby Tables.