Is the aforementioned method for marriage, which includes praying two rak'ahs, reciting Surat Al-Inshirah 314 times, and a specific supplication on Friday night, correct and proven by a Prophetic hadith, or is it an innovation (bid'ah)?

There is no Hadith or narration indicating the method mentioned. However, reading the Quran with the intention of performing Ruqyah or fulfilling a permissible need is legislated, based on the Prophet's (peace be upon him) saying: "There is no harm in Ruqyah as long as it does not involve polytheism," and "Whoever recites the Quran, let him ask Allah by it." The general principle is the permissibility of reciting a Surah and then supplicating afterwards. However, adhering to its repetition a specific number of times or at a specific time, and believing this to be a Sunnah, is an additive innovation (bid‘ah idafiyyah). This is because Al-Shatibi defined an additive innovation as adherence to specific forms and manners at specific times for which such specification was not mentioned in the Sharia.