The gravitational influence of the gas giants (plus general-relativistic effects) are known to cause the apses of the Earth's orbit to precess. For an illustration of that process, see this image. Now, it's clear that apsidal precession combined with changes in orbital eccentricity will affect climate - at minimum, this changes the mean (year-averaged) Earth-Sun distance, which one would of course expect to have an effect on climate.
I'm not really following how apsidal precession alone is supposed to affect climate, though. Let us suppose that the Earth's "rotational parameters" (e.g. axial tilt) are held constant, and that the orbit of the Earth is static except for apsidal precession (i.e. no changes in eccentricity, no orbital decay, etc.).
In this case, the total insolation experienced by the Earth over the course of a year should be independent of how much the apses have precessed - indeed, an observer looking at the Sun-Earth system from above the ecliptic plane should not be able to distinguish between, say, a 5-degree precession of the apses and a 5-degree rotation of the observer in the opposite direction. Since the latter cannot affect the Earth's climate, we should neither expect the former to.
Clearly, I'm making some sort of mistake here, but I'm not sure where exactly my thinking has gone wrong. Where is my error?