How can I achieve the Fox codename?

Solution 1:

So, I started killing one enemy per run through, since that loses the Foxhound requirement - but then I started getting Octopus again.

Well, stop tranqing people so much. It's really that simple. The game has a codename "priority" system. FOXHOUND is at the top because it is the best, so the game will always award you FOXHOUND if you meet the criteria. I would not have thought that Fox is at the same priority as Octopus, but it's not impossible.

Try to avoid using the tranquilizer and kill one or two enemies. Decoys are really effective if you have not been using those. Neutralize as few enemies as possible while still getting one or two to negate the FOXHOUND codename.