I encounter the exact same ColorChecker DCP problem as you did with Fuji pixel shift DNGs in Camera RAW workflow. This appears to be a known bug of Fuji DNG. But it's been a year since your post and I see no one answered your question so far, I wonder why. I assume you probably have already found a solution for this, if so, do you care to share your insight?
I'm also curious about Adobe DNG Profile Editor (ADPE) where you mentioned you were able to export a DCP but couldn't apply its settings to your DNG. ADPE doesn't export DNG file, that's kinda expected. But what about using a pixel shift "ColorChecker" DNG image to generate a profile from ADPE (instead of from ColorCkecker's own Camera Calibration)? And then export this custom DCP to Adobe Camera RAW for pixel shift DNG editing, wouldn't that be more feasible?