Why didn't the Muslim rulers include the Hatim of Ismail (Hijr Ismail) within the Kaaba when it was rebuilt, given the availability of funds, especially since it is part of it and insufficient funds previously prevented its completion?

The Hijr is part of the Kaaba. Aisha asked the Prophet, peace and blessings be upon him, about it, and he replied that his people lacked sufficient funds, and that were it not for her recent conversion from Jahiliyyah, he would have incorporated the Hijr into the House and leveled its door to the ground. Ibn Al-Zubayr built it according to the Prophet's wishes. However, Al-Hajjaj later restored it to its original construction, by order of Abd Al-Malik ibn Marwan, who later regretted it. Al-Mahdi intended to restore it to Ibn Al-Zubayr's construction, but Imam Malik dissuaded him, fearing that caliphs would then treat it as a plaything.