I'll go with an Equivalence Principle argument. For a model system, consider a test particle in a highly elliptical orbit around a neutron star; the particle will pass through regions of greatly different field strength.
But it feels no force as it "falls" around the star. Per the Equivalence Principle, at each point there is a locally inertial coordinate system (the freely falling frame) in which the laws of motion are the same as for special relativity, with no gravity (and hence no force).
Update: I want to poke at a couple things:
1) Your intuition that there is no perception of acceleration while falling in a locally constant gravitational field is actually your internalization of the Equivalence Principle, which implies that, in a gravitational field, all matter falls with the same acceleration. Since Galileo, people have been testing this proposition on various substances, and have never detected a difference (once other effects, such as air resistance, have been accounted for).
Instead of a "test particle" in my model system, suppose I substituted a block of crystalline copper, assumed small enough that tidal forces are undetectable. Crystalline copper comprises a matrix of ion cores immersed in a sea of conduction electrons. If electrons fell at a different rate than the nucleons of the ion cores, the conduction electrons would get pushed to one side of the block, and a voltage difference across the block would be created. That doesn't happen (to the best of our knowledge).
2) Similarly, arguing that acceleration cannot be perceived because there's nothing to compare with is only valid because of the Equivalence Principle. If different components of our bodies accelerated differently, there would in fact be differences to compare, and you could (at least in principle) perceive them as differential stresses.