Is dedicating the last hour of Friday to prolonging prostration (sujood) and supplication (du'a) with the intention of seeking the hour of responsiveness (sa'at al-ijabah) considered an innovation (bid'ah)?

The exact time of "the hour of response" on Friday is a matter of scholarly disagreement, with the strongest opinions being: 1. Between the Imam's sitting on the minbar and the end of the prayer: This is based on the hadith of Abu Musa al-Ash'ari: "It is between the Imam's sitting [on the minbar] until the prayer is concluded." This opinion was favored by many, such as Muslim and Al-Nawawi. 2. After the Asr prayer until sunset: This is based on the hadith of Jabir: "Seek it in the last hour after Asr." This was favored by Ahmad, Ibn Abd al-Barr, and many of the early Muslims (Salaf).

Both hours are times for intensified supplication. The hour of prayer itself contains many opportunities for supplication (between the Adhan and the sermon, between the two parts of the sermon, during prostration, and after the final Tashahhud).

As for the Prophet's (peace be upon him) statement, "while he is standing and praying," it has two possibilities: 1. Waiting for the prayer: For one who is waiting for the prayer is considered as if they are praying. 2. Supplication : The word "prayer" in the Arabic language can also mean 'Dua', and "standing" can mean persistent in supplication.

To catch the hour of response after Asr: 1. Remain in the mosque supplicating until sunset. 2. Go to the mosque a period before Maghrib to supplicate. 3. Supplicate anywhere (e.g., at home) during the last hour of Asr.

Your assumption that "while he is standing and praying" means a prayer with bowing and prostrating is incorrect. Also, prolonging the second prostration in the Tahiyat al-Masjid (salutation of the mosque) prayer goes against the Prophet's (peace be upon him) guidance.