You always can shift clock automatically (on Linux).

Obviously a problem of this sort is not due to the mere numerical value of the time but rather some daily scheduled process that runs around 1am (perhaps a database backup) which interferes with the web app in some way. This sort of thing can be simulated but it's a non-trivial task and it's probably better to apply those efforts in other ways (by diagnosing the problem, putting in place a performance monitoring system, buying more hardware, etc.)