I'll post this answer CW, because I don't have time to work out the details. Hopefully, someone can fill these in or shoot down the strategy.

On wikipedia, there is a cute remark about characterizing meagre sets using a Banach-Mazur game. Basically, you have two players who take turns to build a nested sequence of open sets $O_n$. If $U=\cap_{n=1}^\infty O_n$ is the resulting intersection, one of the players aims to have $U \cap X =\varnothing$ and the other player aims to have a point from $X$ in $U$.

Then, $X$ is meagre iff the player who wants the empty intersection has a winning strategy.

Couldn't this characterization be used in this problem? As I mentioned, I haven't been able to make the details work right...