Pardon the radio silence. Yes, I have this set to print on the second extruder, it's what I had TPE loaded in. It's a problem where if your extruder set to print the part has a print temp of 0, Cura crashes. There is nothing inside the model, either @thopiekar. I've heard that Cura doesn't handle small perimeters well, so this might be the issue. Not sure why @AbeFM has had such better results, an imperfect mesh shouldn't have caused all those rings. Can't see why my settings were causing this, either. Regardless, I definitely think that this is a Cura issue, and that the dev team should look into solving this.
@thopiekar, I don't have the model on hand right now, hopefully the attached screenshot of the sliced model will do.
EDIT:
Also of note, the rings are almost entirely removed when the model is oriented upside down. I've printed it this way (with supports) and it came out nicely, aside from inevitable TPE stringing. No change in settings other than supports activation and dual extrusion tower. Odd.