So there are so many solutions for general relativity - then how does one "choose" the solution that is right one? By checking with observation? (though I also know that it is currently unknown which one is the correct solution.)
Generally speaking we start with a known stress-energy tensor and boundary conditions and look for solutions for the curvature. When doing this we're not usually overloaded with possible solutions, and it's normally pretty obvious which solutions are physically relevant.
Where multiple physically relevant solutions exist we select the appropriate one by comparing with experiment. For example we select the FLRW rather than the Gödel metric because experiment suggests the universe has no net rotation. Likewise the value of $\Lambda$ has to be fixed by experiment, as was done in 1998 by Perlmutter and Riese's groups.