In order for the answer to become clear, several points must be considered:

1- Bid’ah means for something which isn't part of religion to be considered part of it. It has also been proven (in other discussions) that the prophet’s (pbuh) and imams’ (as) actions and sayings are all part of and constitute religion. Keeping this in mind, what you claimed about how adding the household of the prophet (pbuh) to the salawat is bid’ah is false, because even if we assume that it hasn’t been mentioned in the Quran as you put it, it indeed has been mentioned in hadiths from the prophet (pbuh) and the imams (as), making it part of religion. What jurisprudents mean when they say that adding “اشهد ان علیا ولی الله” with the intention of it being part of the adhan is bid’ah, is that they haven’t found any Quranic or hadithic proof that shows this to be part of the adhan (thus, they say since there is no religious proof that it is part of the adhan, we can't say it is, or add it with the intention of it being part of it). To make it short, it is Islam (the Quran, traditions and other Islamic sources) as a whole that determines whether or not something is part of religion or not, not only the Quran; the Quran is one reference for this, but not the only one.

2- All of the different obligations and duties Allah (swt) has instated for us, such as taharah, prayer, fasting, Hajj, jihad, khums, zakat, etc., have all been mentioned in the Quran; the only thing is that they have only been pointed to, without many of their details being explained. Only some of the conditions and parts of these obligations have been expressed in a general way. For instance, the necessity of prayer has been mentioned in the Quran, also some of its conditions have also been mentioned, but what should be said in the ruku and sujud of the prayer, recitation, the tasbihat arba’ah, how to make wudhu, ghusl and tayammum etc. haven’t been explained in detail at all.

The method Muslims used in the advent of Islam was to refer to the prophet (pbuh) and ask him in order to learn of these details and put them to practice.

This was the case with the ‘salawat’ at that time as well. After the verse “ان الله و ملائکته یصلون علی النبی یا ایها الذین آمنوا صلوا علیه و سلموا تسلیما”[1] was revealed, the Muslims went to the prophet (pbuh) and asked him how they were supposed to send salawat upon the prophet (pbuh), and as has been narrated in numerous hadiths by both Shia and Sunni sources (mutawatir), the prophet (pbuh) answered: “Say: “اللهم صل علی محمد و آل محمد کما صلیت علی ابراهیم و...”[2]; this is how the Muslims learned to send salawats upon the prophet (pbuh).[3]

In other words, Allah (swt) has introduced his prophet (pbuh) as the true interpreter and clarifier of the Quran[4], and he (the prophet) has interpreted the abovementioned verse[5] by saying that what is meant by reciting salawat is to say: “اللهم صل علی محمد و آل محمد”. Conclusion: Adding the household of the prophet (pbuh) to the salawat is understood from the Quran itself.

3- In addition to its ‘apparent’ meaning and surface, there are deeper and more profound meanings that only the infallibles are aware and can inform us of.[6] Therefore, it can be said that those hadiths that tell us to add the household of the prophet (pbuh), are actually informing us of the deeper meaning of the verse. In other words, they are telling us that although the meaning of ‘prophet’ or ‘النبی’ only refers to the Prophet Muhammad (pbuh) in normal circumstances, but here covers the Ahlul-Bayt as well, which is a deeper meaning than the normal one.

4- It can be inferred from Quranic verses that Allah (swt) looks at the prophet (pbuh) and his household as one truth and reality. For instance, in the verse of ‘Purification’ or ‘تطهیر’, Allah (swt) says: “انما یرید الله لیذهب عنکم الرجس اهل البیت و یطهرکم تطهیرا”[7]; in this verse Allah (swt) has addressed them as one reality.

Another example would be the verse of Mawaddah or love: “قل لا اسئلکم علیه اجرا الا المودة فی القربی”[8]; when Allah (swt) refers to the love of the household of the prophet (pbuh) as the reward for prophet’s (pbuh) endeavors and efforts, it comes to show that the two never part in His eyes.

Another example is the verse of Mubahalah or malediction: “قل تعالوا ندعوا ابنائنا و ابنائکم و نسائنا و نسائکم و انفسنا و انفسکم...”[9]. As you can see, in this verse, Imam Ali (as) has been regarded as the prophet’s self, and the verse tells us that being the victor in this mubahalah is totally contingent upon all of those mentioned being present.

5- There are various hadiths that tell us that the prophet (pbuh) and his progeny are one light. Take this example: The prophet (pbuh) of Islam says: “I was created from God’s light, and my Ahlul-Bayt were created from my light.”[10]

6- Arefs refer to the prophet (pbuh) and his progeny as “The Muhammadian Truth” or “The First Emanation” which is one reality with different manifestations.

7- In the eyes of normal people, those who share the same beliefs and ideas and are in no conflict with each other, neither in practice nor in thoughts and belief, are all considered one and in other words, people look at them with the same view.

Conclusion: Not only isn't adding the household of the prophet (pbuh) to the salawat not bid’ah, but is in total accordance with the Quran and hadiths and even the view of normal people. Sending salawat upon the prophet (pbuh) is the same as sending it upon the Ahlul-Bayt and both are manifestations of one reality. The last point is that reciting salawat isn't wajib absolutely and rather becomes wajib only in certain cases, like in the tashahud of prayers.[11] Of course, when the salawat does become wajib, it must be recited with the “آل محمد”, and in the cases in which it is mustahabb, adding “آل محمد” will also be mustahabb.