The modern coordinate-indepenent definition of asymptotic flatness was introduced by Geroch in 1972. You can find presentations in Wald 1984 and Townsend 1997. The definition is in terms of existence of something -- a certain type of conformal compactification -- and in many cases such as the Schwarzschild metric, it's straightforward to think up a construction that proves existence. But to prove that a spacetime is not asymptotically flat, you have to prove the nonexistence of any such compactification, which seems harder.
What techniques could be used to prove that a spacetime is not asymptotically flat?
The only example of such a technique that I've been able to think of is the following. Suppose we want to prove that an FLRW spacetime is not asymptotically flat. Wald gives a theorem by Ashtekar and Hansen that says that if a spacelike submanifold contains $i^0$, it admits a metric that is nearly Euclidean, so that, e.g., the spatial Ricci tensor falls off like $O(1/r^3)$. This implies that there can't be any lower bound on the Ricci tensor, but an FLRW spacetime can have such a lower bound on a constant-time slice, since such a slice has constant spatial curvature.
Wald, General Relativity