Is the prayer of one who intends the obligatory prayer (Fajr or Dhuhr) with the takbīrat al-iḥrām valid, compared to it?

Intention (niyyah) is a pillar of prayer. It is best that it be simultaneous with the takbīrat al-iḥrām (the opening Allahu Akbar), but there is no harm if it precedes it by a short time. Some scholars have stipulated that the intention must be simultaneous with the takbīrat al-iḥrām. Your prayer is valid, and what you are doing is the best and most cautious approach.