You can join tables in different databases if they are on the same server. Also consider using placeholders (?) in the queries. For example

#!/usr/bin/perl
use strict;
use warnings;
use DBI;

my $dbh = DBI->connect('dbi:mysql:alicenet','user','password', 
 {mysql_enable_utf8 => 1,RaiseError => 1,PrintError => 1})           
 or die "Connection Error: $DBI::errstr\n";

my $where = 'WHERE M.failed < 3
             AND M.lasttry < (NOW() - INTERVAL 90 MINUTE)';
           
my $sql   = "SELECT COUNT(id) 
             FROM mailcue AS M
             $where";
 
#count the messages
my ($count) =  $dbh->selectrow_array($sql);

#max number of cued messages to fetch
my $max = ($count < 180)? 30 : int($count/6);
 
$sql = "SELECT U.email,M.subject,M.message,M.id,M.failed 
        FROM mailcue as M
        LEFT JOIN anetauth.userauth as U
             ON   U.id = M.userid
        $where
        ORDER BY M.id
        LIMIT ?";

my $sql_fail = "UPDATE mailcue 
                SET lasttry = NOW(),failed = ?
                WHERE id = ?";
my $sth_fail = $dbh->prepare($sql_fail);        

my $sql_ok   = 'DELETE FROM mailcue 
                WHERE id = ?';
my $sth_ok   = $dbh->prepare($sql_ok);        

my $ar = $dbh->selectall_arrayref($sql,undef,$max);
for (@$ar){
  my ($email_addr,$subject,$message,$id,$failed) = @$_;
  # replace print with email create and send
  print "
  --------------------
  to      = $email_addr
  subject = $subject
  body    = $message
  
  id      = $id
  failed  = $failed
  ==================";
  my $response = 'fail'; # test
  if ($response eq ''){  # change to suit
    $sth_ok->execute($id);
  } else {
    $sth_fail->execute($failed+1,$id);
  }
}
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