OK, we really need a repeatable test if possible. Perhaps a pgbench run with lots of concurrent runs of a some very long query would do the trick.
OK, I can do it but ...strangely that error usually occurs at random times, sometimes at 08:00, sometimes at 19:00, and it's busier between 10:00 and 16:00. If I cron some of those queries to run every second is enough ? What exactly do you expect to see on log files ?