If the text of Srimad Bhagavatam is dated around the year 1000 AD, how can you appreciate the prediction of the birth of Lord Buddha? And an additional question: since the advent of Lord Rama was before that of Lord Krishna, would this not affect the order of chanting the Mahamantra? (i.e. Why does the Mahamantra not begin with Hare Rama instead of Hare Krishna?

The dates given by scholars of the Bhagavatam are not concrete. These are speculations. In any case, a famous aspect of the Puranas, in general, is containing texts and stories from many different periods. The prediction of Buddha is accepted as such.

The order of the names of the Mahamantra is not chronological, but in the order of spiritual potency. Also, Rama can indicate Lord Balarama.