Since I started focusing on strongman as opposed to powerlifting, OH has been my focus, so I've figured out pretty well what works for me. Here's what I've learned through experiment and from other competitors:
When my push press goes up, I can usually hit a military press PR a few weeks after switching back to a regular strict press. The reverse is not true.
Also, the last 2 contests I did, I focused on my military in the 4 weeks prior to my contest, and even though I felt stronger, I didn't hit a good PR in contest. I figure this is from not being used to using drive.
3rd, I get (and others have confirmed this) better results with a higher rep range on military presses than on push press.
I tried rotating them as main movements for a while (3-4 week cycles, push press, btn push press, military) but didn't progress as much as I should have. Now, I'll be doing some kind of push press for my main exercise, but following it with a military press or strict press variation. Most of the strongman competitors I know do a push press or jerk as their main overhead movement, but somewhere in the week keep a strict press variation for more reps (accessory). They also tend to be massively wide. This falls in line with the 3 things I posted about.
You, however, might not be able to do this because of your back injury. Something to try would be to do your sets of military press with w/e weight, and then afterwards rep it out using some light leg drive. The weight would be small enough that the risk of reinjury would be small, but there would still be overload.
Damn that was long.