In prayer, after Tashahud, we do Salawat on the prophet (s) and his family (as). Is it allowed to extend this salawat and say the Salawat al-Ibrahimiyya in prayer? ("allahuma salli ‘ala Muhammad wa aali Muhammad, kama salayta ‘ala Ibrahima wa aali Ibrahim…")

No we are not allowed to add anything that Salawat.