How do we reconcile the miracle of the splitting of the moon in the hadith which narrates that the people of Mecca asked for a sign, and the Prophet showed them the moon split in two, with the Quranic verse that alludes to the fact that Allah would not send physical signs due to the denial of the previous peoples, and that the Quran is the only miracle for the challenge?

There is no contradiction between the miracle of the splitting of the moon and the noble verse, because the verses upon whose denial destruction was predicated are those that the polytheists specifically requested. As for the splitting of the moon, they did not request it. And Allah delayed the punishment from the disbelievers of Quraysh, knowing that among them were those who would believe or be born believers, after they requested specific signs, such as turning al-Safa into gold. Then Gabriel descended with the choice between immediate destruction or postponement, and the Prophet, peace and blessings be upon him, chose postponement. As for the splitting of the moon, it was a sign they did not specifically request; rather, they requested a sign without specifying it, and so the moon split.