Not to derail your thread and I’m certain you have already checked them out, but take a good hard look at the SX3, shootoff, or perfex from Stan. I like Stan better than Carter myself. Generally have more adjustability with the thumb barrel and the trigger mechanism itself is every bit as good if not better than a Carter.
That being said, Carters are excellent as well. And to actually answer your question, I’d lean toward a thumb trigger for hunting. Out of the two you listed I’d go with the wise choice. I like the index finger hole.