You certainly can play the game with rationals, and Bob can win!
The idea is that if Alice's sequence looks to be converging to some rational number α, well then α came up in Bob's enumeration of all rationals and was (by hypothesis) legal to play. So Bob played it, and then played something smaller next move! So Alice's sequence cannot converge to α.
The idea is that if Alice's sequence looks to be converging to some rational number α, well then α came up in Bob's enumeration of all rationals and was (by hypothesis) legal to play. So Bob played it, and then played something smaller next move! So Alice's sequence cannot converge to α.