All-In or Split the Stake to Double Up

You have $\$1$ and want to double it to $\$2$ on a subfair even-money bet that wins with probability $p=0.45$ (and you quit forever once you either reach $\$2$ or hit $\$0$). Compare two plans: (A) bet the whole $\$1$ in one shot; (B) bet $\$0.50$ each round until you reach $\$2$ or go broke. Which gives the higher probability of doubling, and what are the two probabilities?