Can a Muslim combine committing a grave sin with consistently performing the Dawn prayer (Salat al-Fajr)?

There is no impediment, rationally or legally, for a person to be diligent in performing Fajr prayer (or the five daily prayers) while simultaneously committing a major sin. This is observed and palpable, and the one who commits a major sin will be under Allah's will in the Hereafter. For the people of Tawhid (monotheism), even if they commit many sins, they do not disbelieve by them as long as they adhere to Tawhid, and forgiveness is hoped for them, in accordance with His, the Exalted, saying: {And He forgives what is less than that to whom He wills}.