My problem was that the camera wasn't letting me choose H or H+ in the mechanical shutter mode. It must have been a setting I choose in another menu that blocked the choice. I just solved the problem by doing a complete reset.
Choosing Highlight Tone Priority (HTP) disables the option to shoot at ISO 100. Turning on HDR PQ in stills mode (or video) automatically turns on HTP, but you can turn it off and still use HDR PQ, as far as I know.
I am guessing you had the Auto Level feature turned on. When that is turned on, you cannot choose H or H+ in mechanical shutter mode and cannot choose EFCS at all. You can choose electronic shutter and still get H and H+ though.
Adding to your llist - turning on Auto Level (sensor rotation to straighten horizons) also stops H/H+ being available in mechanical shutter mode, completely disables EFCS and does allow H/H+ in electronic shutter mode.
