The very short answer is that he was already sharing his timer between himself and the little girl. He wasn't sure how much time he had left, courtesy of Miss Flitcroft's timer (possibly only seconds) and grabbed for the first weapon he could reach.
Renata Flitcroft was able to give Bill Door some of her time, but given that she was also almost dead (of a heart attack) her "borrowed time" didn't seem sufficient for him to do anything than try to attack the new Death with the old scythe he'd used for the harvest.
The fancy scythe he'd crafted was all the way on the other side of town. The working scythe was directly behind him. If he'd gone for the other scythe, his extra time may simply have run out, leaving him in an even worse position:
TONIGHT. I CANNOT BE EXACT. TWO PEOPLE ARE LIVING ON THE SAME TIMER.
IT MAKES THINGS UNCERTAIN.
Between her hands, in the air in front of her hovered the faint
outline of a lifetimer, its sand pouring away in a torrent...
...But Bill Door was already rising and unfolding like the wrath of
kings. He reached behind him, growling, living on loaned time, and
his hands closed around the harvest scythe.
Ultimately it didn't really matter what weapon he used. His sense of purpose gave the blade a (metaphorical) edge that couldn't be beaten.
The crowned Death saw it coming and raised its own weapon but there
was very possibly nothing in the world that would stop the worn blade
as it snarled through the air, rage and vengeance giving it an edge
beyond any definition of sharpness. It passed through the metal