So I tried using the QPcalibration software to create an ICC profile for use in Capture One. Holy cow, I got a super flat and color accurate image (when using Linear Response Curve) straight away. Just a little tweak with the skin tone tool, and I got significantly better results than I ever got with DCP profiles in Adobe.
I used Large Gamut, Low Profile Curve, and the High Smoothing settings in QPcalibration. The QP profile was still very contrasty in Lightroom, even when I dialed the Contrast down by -80.
Here are the results, Lr first, then C1 (the differences are greater on my calibrated editing screen than these small sRGB files will yield, but you can still see the contrast differences in her dress, shoes and face):