Al-jawab billahi at-taufeeq (the answer with Allah's guidance)
There is no Shar’ee basis to use the number 786. It is not preferable either.
Also, 786 in no way represent ‘Bismillaah al-Rahmaan al-Raheem’ and is not the Sunnat way.
CAN NUMBER 786 REPLACE THE HOLY NAME OF GOD?
The innovation of writing '786' replacing 'Bismillaah al-Rahmaan al-Raheem' has been adopted for a long time and the majority of the Ummah is still indulged in it inadvertently.
Apart from the common folk, the scholars also heed no attention towards it and to avoid disrespect to the Holy Words they use it in their letters and documents. They adopt it as 'correct' and 'better' way to invite Allah's blessings and have also started replacing the Holy Words by this number on their houses, offices, buildings, etc. Unfortunately, this tendency is gradually gaining momentum.
But, do we see this number instead of the Holy Words in the Holy Qur'aan? Or, can we write it there as well? can we remove 'Bismillaah al-Rahmaan al-Raheem' from the top of Surah al-Faatiha and replace it with the number 786? Obviously not.
If we study the Qur'aan, we see it carries the holy words in a letter from Prophet Solomon (pbuh) to the Queen of Sheeba - Bilqis - who was an infidel at that time. Even Prophet Muhammad (pbuh) in his letters to different heads of states and governments, used the holy words at the top.
Therefore, whosoever uses 786 with the intention to obtain Allah's blessings, is a misguided person and any attempt to justify it, is ignorance.
More astonishing is that fact that '786' is an aggregation of the numbers of Hindu 'Lord Hari Krishna'.
h-5, r-200, r-10, k-20, r-200, sh-300, n-50, a-1 = Aggregate of 786
Thus, the aggregate number of these letters (Hari Krishna) equals 786. This is also the case of 'Bismillaah al-Rahmaan al-Raheem'. Therefore, it is necessary to avoid using this number to avoid the danger of being indulging in infidelity.
Islam's foundation is laid on the belief in Tawheed (oneness of God). If we associate anyone with Allah's exalted names orally or practically, we would be committing infidelity which is an unpardonable sin.
The Qur'aan warns us, one who finds a rival against Allah, Allah will never allow him to enter paradise, and his abode is the hell.
The letters by the Prophet (pbuh) to non-Muslim Kings and chiefs bear the holy words 'Bismillaah al-Rahmaan al-Raheem'. This was also the case of his noble companions. Did the Prophet (pbuh) and his companions respect the Holy Words less than we do? Was the verse, 'Today I completed your religion for you', revealed to the Prophet (pbuh) or to the socalled scholars of our age who are all bent to amend the Divine principles.
Now, when it became known that '786' is written in place of Hindu, 'Lord Krishna', and has no significance in the Holy Qur'aan and Sunnah, it is obviously an innovation and goes against the Divine doctrine of 'Obey Allah and His Prophet'.
Similarly, some Muslims replace Allah's name by the number '66' and the Prophet's name by '92'. But if one ask, the number '420' is used against somebody's name and he is called '420'. Can he bear such an insult?
Calling someone by a nick name is a sin. Allah Ta'ala says, 'Don't call one another by nick names'. If Allah prohibits the use of nicknames for fellow human beings, how can He allow the use of such insulting replacement number for Himself and His Prophet (pbuh).
'Bismillaah al-Rahmaan al-Raheem' great significance in the Qur'aan and Muslims are ordered to begin any work but in the names Allah, not in the number of '786' or any other substitution.