To date, alternating co-polymers based on electron-rich and electron-poor units are the most attractive materials to control functionality of organic semiconductor layers in which ultrafast excited-state processes play a key role. We present a computational study of the photoinduced excited-state dynamics of the 4-(2-thienyl)-2,1,3-benzothiadiazole (BT-1T) molecule, which is a common building block in the backbone of π-conjugated polymers used for organic electronics. In contrast to homo-polymer materials, such as oligothiophene, BT-1T has two non-identical units, namely, thiophene and benzothiadiazole, making it attractive for intramolecular charge transfer studies. To gain a thorough understanding of the coupling of excited-state dynamics with nuclear motion, we consider a scenario based on femtosecond time-resolved x-ray absorption spectroscopy using an x-ray free-electron laser in combination with a synchronized ultraviolet femtosecond laser. Using Tully's fewest switches surface hopping approach in combination with excited-state calculations at the level of configuration interaction singles, we calculate the gas-phase x-ray absorption spectrum at the carbon and nitrogen K edges as a function of time after excitation to the lowest electronically excited state. The results of our time-resolved calculations exhibit the charge transfer driven by non-Born-Oppenheimer physics from the benzothiadiazole to thiophene units during relaxation to the ground state. Furthermore, our ab initio molecular dynamics simulations indicate that the excited-state relaxation processes involve bond elongation in the benzothiadiazole unit as well as thiophene ring puckering at a time scale of 100 fs. We show that these dynamical trends can be identified from the time-dependent x-ray absorption spectrum.