Is it a Bid‘at to mumble or say the Nīyyah of Salāh in a very low tone which may not be audible to others ? I read somewhere that Nīyyah should only be in the heart.

The heart's intention to do something is called Nīyyah. It is therefore something which pertains to the heart only and is not required to be pronounced. One should not utter the Nīyyah of Salāh by mouth since the Prophet (sws) never did so. There is not a single directive of the Prophet (sws) which directs us to pronounce the Nīyyah in the manner the Hanafites of this age generally do. The principle is that in all matters of worship, the Prophet (sws) must be followed exactly.

This is what the celebrated scholar Ibn Qayyim says while describing the prayer of the Prophet (sws):

When the Prophet (sws) would stand up for the prayer, he would say: Allāhu Akbar. He would never say anything before this nor would he ever pronounce his Nīyyah, nor would he say “I am offering this particular prayer consisting of four Rak‘ats for Allah facing the Qiblah singularly or as an imam”, nor would he say: “[I am offering this prayer] within the stipulated time or after the stipulated time or that these are the obligatory Rak‘ats of this time.” These ten innovations have never been narrated by anyone – not a single word of them -- either as a Sahīh or a Da‘īf or a Musnad or a Mursal narration. There is nothing in this regard that has been narrated from the Companions of the Prophet and no one from among the Taba‘īn or the four imams ever regarded it as desirable. Actually, some scholars of the later period have misinterpreted these words of Shāf`i‘ī (rta): “It [the prayer] is not like the fasts. No one can begin it unless he pronounces [some words]”. They thought that this meant the pronunciation of the Nīyyat, whereas Shāf‘i‘ī  actually meant the pronunciation of Allāhu Akbar, which are the words that commence the prayer. For how can Shāf‘i‘ī regard something as desirable which the Prophet never did even in one single prayer and which no one from anyone of his Caliphs and Companions did either. This is the guidance [they have provided us] and this is their practice. If we had found that a single word had been narrated by them in this regard, we would have accepted it and faced it with submission and compliance for no guidance is more complete than their guidance and there is no established practice [in religion] except that which has been received from the Prophet – the law giver. (Zād al-Ma‘ād, Vol 1, p. 201)