Does the Srimad Bhagavatam comment on its time of origin?
Yes, the somewhat similar questions are clarified in the Bhagavatam: Chapter 4:
Text 3: In what period and at what place was this first begun, and why was this taken up? From where did Kṛṣṇa-dvaipāyana Vyāsa, the great sage, get the inspiration to compile this literature?
After compiling Vedas and Mahabharata, Vyasadeva was not satisfied. On the banks of the river Sarasvatī, Vyasadeva got inspiration from sage Narada to write Śrīmad-Bhāgavatam. So, it's time of origin is after Mahabharata war (at the conjunction time of Dwapara and Kaliyuga).