I'd cross off
Ernest because it has the same old clunky feel as
Arnold if it were a first name, which obviously you can't get around having in there. So putting the two together feels like overkill.
Gideon is a lot more fresh and "modern" feeling, similar to
Theodora, so I'd go for that of the two. I don't think it would get shortened, I've never known one who went by anything but
Gideon. Plus it's not really that hard to correct people if they did try to shorten it without asking.