What is the ruling on not saying "Bismillah" over the sacrificial animal, especially when the one slaughtering it does not pray?

The sacrifice of one who abandons prayer is absolutely impermissible. As for the naming (mentioning Allah's name) upon the sacrifice, scholars have differed on this matter, holding three opinions:

1. It is only recommended (Shafi'i school). 2. It is a condition, but it is waived if forgotten (Hanafi, Maliki, and Hanbali schools). 3. It is an indispensable condition that is never waived (Zahiri school, a narration from Malik and Ahmad, and the choice of Ibn Taymiyyah and Ibn Uthaymeen), who base their argument on the generality of Allah’s statement: (And do not eat of that upon which the Name of Allah has not been mentioned) and the Prophet's (peace be upon him) saying: (Whatever causes the blood to flow and the Name of Allah is mentioned upon it, then eat).

Therefore, only those who perform prayers should slaughter the sacrificial animal (Qurbani) and other animals. It is a condition to mention "Bismillah" (in the Name of Allah) at the time of slaughter, and it is also recommended to say "Allahu Akbar" (Allah is the Greatest).