Mike, This is the first time you affirmed the BR had acquired first base. If he did, you are correct, third base would be the award.
But you said something about the BR had attained first base when the "muff" occurred. You also said the runner was between first and second base when the ball went out of play. Neither of those two comments have anything to do with placing the runner. Time of throw or time of pitch is what we are concerned with.
In my experience, a decently trained infielder would never release a throw if the BR had already reached to base.