There is no hadith attributed to the Prophet ﷺ about takbīrat during the time of Eid al-Adha apart from:
حدثنا عفان، حدثنا أبو عوانة، حدثنا يزيد بن أبي زياد، عن مجاهد، عن ابن عمر، عن النبي ﷺ قال: ما من أيام أعظم عند الله، ولا أحب إليه من العمل فيهن، من هذه الأيام العشر، فأكثروا فيهن من التهليل والتكبير والتحميد
— NOTE: My own translation, so treat with care:
Narrated by Ibn 'Umar that the Prophet ﷺ said: There are no days in which righteous deeds are more beloved to Allah than those ten days, so increase [your] utterances of profession of faith [tahlīl, or saying La ilāha illa Allah), utterances of His Greatness (takbīr, or saying Allahu Akbar), and utterances of praise of Him (tahmīd, or saying Al-hamdu li Allah).
— Musnad Ahmad, Hadith 5294
Apart from that, there are different opinions of how many takbīrat one can (one or three), the words to use, and the time of doing them (see in Arabic only Sharh al-Zurqānī 'Ala Muatta' Malik, Arabic: شرح الزرقاني على موطأ الإمام مالك) based on what some of the companions practiced. So doing takbīrat is not bid'ah, but there is nothing attributed to the Prophet ﷺ that restricts takbīrat to only after prayers, or to limit them in number.