...but I'm concerned that the pitch down may suggest that I have some kind of airplane rigging problem, or the GI 275 is out of calibraion.
Another vote that it is much more likely the GI-275 just needs to be recalibrated/have the offset readjusted. A true aerodynamic problem that requires you to be 5° nose down in trimmed level flight would manifest other aircraft control issues that should be pretty obvious, not to mention that your airspeeds should be quite a bit off from expected/published numbers.