So, I want to clarify something, because it will become important in your continuing rehab with this horse.
You do NOT want to “strengthen” the latissimus dorsi.
Glutes, yes, lat dorsi, no way, Jose.
It’s not about being picky about word choice, it’s about what is actually happening to the long muscles of the back when we talk about “building the topline”. I actually really dislike that phrase, because it implies we want to DO something to those muscles, when in reality what we really want is to engage the abdominal muscles. If you say “strengthen the latissimus dorsi”, what you are saying is “cause repetitive contraction in the latissimus dorsi” which is exactly the opposite of what we want, because that’s what a horse with a hollow back does: contracts the topline. We want the horse to RELEASE the topline by engaging the abdominals and flexing at the SI join, using the powerful glutes to create push.
So what you want to do is anything and everything that will encourage this horse to release his back by engaging his core. Exercises to do this include series of poles (not raised) with a shortened distance between them to encourage slow, methodical movement and lift, teaching the horse to step under his body with each hind leg (some call it disengaging the hind end, some call it untracking…you’ll hear a lot of different terms), in-hand lateral work to encourage the aforementioned crossover, turn on the forehand, etc.
When you’re to the riding/lunging stage, the walk and canter are where the most movement in the SI happen, so using those judiciously and minimizing your trot work will be helpful.
The carrot stretch where you ask the horse to bring his head between his knees to reach the treat will also help tremendously, as doing so lifts the entire free span of the back.
Good luck. Sounds like it’s been a long process, but then again, you really can’t take too much time with this sort of thing.