Yes swimming is considered contraindicated for horses with back conditions such as kissing spine because of the protracted contraction of the topline (as they hold their heads high above the water). You had said your vet prescribed swimming though, so presumably your vet thinks that the benefits of swimming (of which there are many) outweigh the risks.
I don’t think you specified what the injury was, so whether it’s the kind of thing that will benefit from aquatreading instead of swimming is something you should ask your vet.
Whether an aquatread is better than swimming for general fitness is something that could probably be researched, but the two are different enough I would want to know that the injury isn’t affected negatively by one or the other. Certainly, an aquatread is better for general fitness than no aquatread.
Because the facility is so far away, and your regular vet doesn’t go that far, I would:
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Find out which vets do go there, and whether there is a very competent sports vet that can take over rechecks (I would assume so, if it’s a rehab facility).
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Decide if you are comfortable spending a few weekends going to check on your horse given that you’ve never used it. I currently drive 90 min to visit my horse at rehab. I did it two weekends in a row when I first moved her to make sure I was happy with the care, then switched to ~ once a month. Not sure if you have that kind of time (if you have a farm at home, I suspect not!), but if you can’t get assurances from someone you know at trust about the care, I agree that I would be skeptical. There are so many different standards of horse care and it can be hard. Then again, I often have to remind myself that, as long as the horse is happy, sometimes it’s ok to live with normal horse keeping standards (versus my own which I constantly have to keep in check since I board 
).
If you’re not comfortable sending him somewhere unknown, and the horse was not happy at option 1 (was it full stall rest, without the run? A run can make a big difference in how confined they feel. I’m assuming he was sedated, e.g., trazadone?), I would keep him at home. If he can have full turnout at your place, and you can hand walk him once a day at a good march and with, say, regular walking over cavaletti poles (assuming that is allowed per vet), with tummy tucks and back lifts, etc, to keep a little bit of fitness, then that would probably be fine. You’d be amazed what 45 minutes a day of a good march with a relaxed neck and swinging back can do! (Plus, maybe there’s a facility a little closer with an aquatread that accepts trailer-ins? In our area there are a few places like that that don’t necessarily do full-on rehabs. Maybe you can trailer out once a week? Just throwing ideas out).
Good luck and hope you have a speedy recovery to FULL and long term soundness!
edit: Maybe I’m a little confused - I thought you said the horse was not happy on stall rest. Was that not at the rehab? If he was happy at rehab option 1, send him back there to keep doing the aquatread (since it sounds like your vet wasn’t explicitly saying “swimming,” but more like “water.” Which frankly I find odd to conflate the two, as they are two very different exercises.
and would be happy to have him back but they don’t offer full turnout, only a stall with a run (which they have available for him) and aquatread not pool. Vet also visits every other week.