One explicit form where you can see the topology of the spatial slices in spherical coordinates as $S^3$ is $r=\sin \chi$.
- These are different spatial slices than in the dS FLRW form!
- The global topological consideration should not be done without understanding the whole analytical extension, the whole of Schw.-dS is larger than simple dS!
I am not aware about any "asymptotically FLRW" set of coordinates describing Schw.-dS but this chart would be very ugly because it wouldn't be stationary or homogeneous/isotropic on the spatial slice. Again I recommend Griffiths & Podolský on references on various coordinate charts.
It may sound pedantic but I would also like to stress that these coordinate charts are not different "solutions" but simply coordinate charts mapping the unique solution in different ways. I assume you know about this fact but seeing how many people nowadays do not understand the fleeting nature of coordinates in relativity, I think one should be careful in the language.