Head of the program that implemented a conceptual design, yes, no question.
> he was the conceptual designer for the bomb itself.
Very questionable.
It took multiple attempts by the MAUD committee to convince both US DoD and US physicists that an atomic bomb was even possible let alone feasible (ie practical to build (just) and affordable (just)).
After ignoring several letters, including one from Einstein, it took a personal visit from the Australian physicist Mark Oliphant to campaign at the DoD and to then walk Oppenheimer through the Frisch–Peierls memorandum (students of Oliphant) and convince Oppenheimer to run the calculations on a compressed spherical ball of uranium isotopes (seperated by a process developed by Oliphant) himself.
It's a rarely taught and oft forgotten piece of nuclear history that the impetus and conceptual design for the Manhatten project originated from non US citizens outside of the USofA.