Is it permissible to delay the Dhuhr prayer until half an hour before the Asr prayer to get enough sleep for someone who cannot sleep after Fajr prayer except after three hours?

There is no sin upon you for intentionally performing the Dhuhr prayer half an hour before its time expires. The important thing is that you have prayed it before the time for the Asr prayer begins. Performing it at the beginning of its time is better, but it is not obligatory.