Is it sufficient to invoke blessings upon the Prophet in a single prostration, and to rely on supplication in the remaining prostrations, given that the Imam rises quickly from prostration and does not allow enough time to combine supplication and invoking blessings upon the Prophet?

Praying upon the Prophet, peace and blessings be upon him, is not a condition for the acceptance of supplication, but it is one of the causes for its acceptance. It is not a condition for your supplication to be accepted that you pray upon the Prophet, peace and blessings be upon him, in any of the prostrations. If you do so in one prostration, or two, or as much as is easy for you, it is good, and it can be done in a concise manner. The matter in this regard is broad. The saying, "Supplication remains suspended between heaven and earth unless prayers are sent upon the Prophet, peace and blessings be upon him," was reported in a tradition (athar) attributed to Umar ibn al-Khattab and was deemed good by Al-Albani. This indicates the desirability of praying upon the Messenger, peace and blessings be upon him, along with the supplication, and not that it is a condition for its acceptance.