Contrary to what people think, Kemal actually kept the position of caliph around after the Grand National Assembly won the Turkish War of Independence, but only because the Caliph was still widely respected by the people. Abulmejid was even elected to the position by the Grand National Assembly., but Kemal could barely tolerate him.
After the GNA elected Abulmejid to the position, Kemal refused to permit him the traditional Ottoman ceremony in which his role would be conferred, saying...
So at the bare minimum, the Calph was only worthwhile as a constitutional monarch. When Abulmejid declared that he desired an increase to his allowance, Kemal was even clearer.
The legal position of Sultan was officially abolished in 1923, but the position of Caliph was retained until 1924, when two Indian brothers representing the Indian-based Khalifat movement started a campaign to preserve the Ottoman caliphate. When this happened, Ataturk finally took the chance and abolished the position of Caliph, officially ending the transition of Turkey from the Ottoman Empire to the Republic of Turkey.
So, in this timeline where the Sultan sides with the GNA, I expect this same tenuous relationship to remain and I severely doubt Kemal would be in the mood to shuffle out Abulmejid for a compliant puppet, since as I just said, he could barely tolerate the Caliphate as it stood. At best, the Caliph would be kept under house arrest and only allowed outside if he agreed to toe the Republic's line.
I don't think so. Some people don't understand what the position of Caliph embodies. The word 'caliphate' means 'successor', meaning that they are the successors of the Prophet Mohammad in leading the Islamic faith, or at least, a faction of that faith. If the caliph is effectively a hostage of the Republican government and cannot exert influence over the Islamic community in a particular country, then it brings shame to the title, which was once held by Mohammad's family. Such shame would not be tolerated by the Islamic community of Turkey.
In my opinion, the position of Caliph would be abolished in August 1944, after the death of Abulmejid II.