Why do Ismailis recognize Ismail as the true Imam after Imam Ja'far al-Sadiq instead of Musa Kazim?
Ismaili Muslims recognize Ismail as the rightful and true Imam after Imam Ja'far al-Sadiq, while the Twelver Shi'a recognize Musa al-Kazim — Ismail's half-brother. The Ismaili position for the Imamat of Ismail is based on the historical evidence presented below as Seven Proofs for the Imamat of Ismail b. Ja'far al-Sadiq. Taken individually or collectively, each of these Seven Proofs demonstrate that the legitimate successor and true Imam after the Imam Ja‘far al-Sadiq is Mawlānā Ismā‘īl ibn Ja‘far and that the Present and Living Imam of all Shi‘i Muslims must be the direct lineal descendant of Isma‘il ibn Ja‘far. (Note: The below is just a summary of the proofs. The fully fleshed out proofs can be found in the Ismaili Gnosis article linked at the bottom)
Proof #1: Imām Ja‘far designated Mawlānā Ismā‘īl as the next Imam by the rule of nass as per Twelver, Ismaili, Sunni and academic sources.
Proof #2: The only way to deny the nass of Mawlānā Isma‘il is through contradictory hadiths presented in later Twelver hadith books.
Proof #3: Isma‘il’s death before Imam Ja‘far is not confirmed and may have been staged to protect him — as he was reportedly seen by eyewitnesses after his alleged death.
Proof #4: Even if Isma‘il had died before his father, the Imamat continued in Isma‘il’s son, Muhammad ibn Isma‘il, whom Isma‘il had appointed as his own successor.
Proof #5: Earliest Shi‘i hadiths lack the mention of Twelve Imams but instead predict exactly the first eighteen Imams in the Isma‘ili lineage of Imamat.
Proof #6: With the exception of the Nizari Ismaili Imamat, all other Shi‘i Imamat lineages have hidden Imams. This contradicts the Qur’anic definition of Imamat which requires the Imam always be present and manifest (mubin) in the world.
Proof #7: Imam Shah Karim al-Husayni Aga Khan IV is the forty-ninth hereditary Imam of Shi‘i Islam in direct, documented, lineal descent from Mawlana Isma‘il ibn Ja‘far al-Sadiq. As the only present (hadir), manifest (mubin) and living (mawjud) hereditary Imam, with a documented and validated lineage, Imam Shah Karim al-Husayni’s very existence is itself confirmation of his Imamat and that of his ancestors.
Details of each of the above proofs can be found here: https://ismailignosis.com/2014/10/02/who-succeeded-imam-jafar-al-sadiq-seven-proofs-for-the-imamat-of-imam-ismail-ibn-jafar/