Answer
In the Name of Allah, Most Gracious, Most Merciful.
All praise and thanks are due to Allah, and peace and blessings be upon His Messenger.
In this fatwa:
Some scholars are of the view that adding “Ar-Rahman Ar-Raheem” after saying Bismillah when making wudu is more perfect, and others are of the view that it is better not to add these words and to limit it to saying “Bismillah.”
Answering your question, the Fatwa Center at Islam Q and A, states:
The format is for the one who wants to do wudu to say “Bismillah.” If he wants to add the words “ar-Rahman ar-Raheem,” there is a difference of scholarly opinion concerning that.
Some scholars are of the view that that is more perfect, and others are of the view that it is better not to add these words and to limit it to saying “Bismillah.”
An-Nawawi (may Allah have mercy on him) said regarding mentioning the name of Allah when doing wudu:
“It should be understood that the more perfect way of mentioning the name of Allah is to say “Bismillah ar-Rahman ar-Raheem”, but if one says “Bismillah” only, then he has attained the virtue of mentioning the name of Allah, and there is no difference of scholarly opinion on that score.”
Shaykh Ad-Dardeer al-Maliki (may Allah have mercy on him) said:
“Mentioning the name of Allah means saying, when washing the hands up to the wrists, “Bismillah.” With regard to adding the words “ar-Rahmaan ar-Raheem,” there is a difference of opinion.” (Ash-Sharh as-Sagheer li Aqrab al-Masalik, 1/122)
Sheikh `Abdullah ibn Jibreen (may Allah have mercy on him) said:
The Prophet (peace be upon him) said: “There is no wudu for one who does not mention the name of Allah when doing it.” The point at which that is to be done is when starting to wash the hands, or when starting to rinse out the mouth. One should say, “Bismillah.” Some of the scholars are of the view that it is obligatory, but it is waived in the case of one who forgets or is unaware of it. As for the one who remembers, he should do that by saying, “Bismillah,” and that is sufficient. If he completes it by saying “Bismillah ar-Rahmaan ar-Raheem”, that is permissible.” (Sharh ‘Umdat al-Ahkam by Ibn Jibreen)
With regard to the view that if someone says “Bismillah ar-Rahman ar-Raheem”, it is not acceptable, we have not come across any of the scholars saying this. Rather their difference of opinion is as to whether it is better to add this phrase or not, and not one of the scholars says that if someone adds it, his wudu is not valid.
Almighty Allah knows best.
Source: https://islamqa.info/en/225088