When we define $\Bbb{CP}^1$ as a complex $1$-manifold, we give it two charts $(U_0,\gamma_0)$ and $(U_1,\gamma_1)$. We also say it has a complex structure $\Sigma$, which is an equivalence class of analytically compatible atlases.
Does this mean that $\Bbb{CP}^1$ has only two charts, or does it have every chart that is compatible with these two charts as well (surely infinitely many)?
It has all the compatible charts - indeed infinitely many, and you can choose the ones you want to use in any particular computation.
The minimum number of charts that will cover the manifold is two. The two you name are the conventional ones.