Which exactly is what general relativity is centrally about and the reason why general relativity is defined in terms of intrinsic properties of the Einstein manifold describing the space of the universe. Intrinsic properties are, colloquially, indeed properties that can be measured "using a ruler that inhabits the same space".
Gauß' Theorema Egregium [1] is an early predecessor of intrinsic properties of general manifolds, and says that the curvature of a surface can indeed be "be determined entirely by measuring angles, distances and their rates of change on a surface, without reference to the particular manner in which the surface is embedded in the ambient 3-dimensional Euclidean space" ([1]).
[1] https://en.wikipedia.org/w/index.php?title=Theorema_Egregium...
In LIGO, we detect the far away distortions propagating through our rulers and the differential effect of those waves.
But Michelson-Morley failed because the ruler distorted synchronously with the signal we tried to measure — since we mistakenly believed EM and matter were different fields.
General-relativistic effects are relevant for the outcome of the Michelson-Morley experiment in the sense that the general theory of relativity is a generalization of the special theory of relativity. :-)
Otherwise: the additional predictions that the general theory of relativity makes about gravity, curvature of space etc. are not relevant for the outcome of the Michelson-Morley experiment.
We do that, routinely: that’s how we do cosmology.
Yea, that is why we are not getting any where...
Which alternative way of doing measurements do you propose? As far as I know, measuring intrinsic properties, i.e. "using a ruler that inhabits the same space" is with our current technology the only possible way of doing measurements.