A cat could reenter the garden and be killed in less than 15 seconds. At the very least, if you continue with the current arrangement, you should not let him roam the garden freely while the cats are outside. Just because you don't see one there when you let him out doesn't mean they will be safe. While the cats are loose outside, you can take him out on leash. You may find a long-line (aka training leads in 15/20/30 feet or more) useful for this.
I don't know what your feelings are about cats as pets, but I don't think it's right to only allow your cats indoors for ten minutes per day. Personally, I don't allow my cats outside at all -- but even if I did, they would have free access to their home at all times. It sounds like you are prioritizing the new dog over the cats' welfare, which I don't think is right. I'm not trying to be belligerent or unhelpful. I'm just genuinely worried about this arrangement. It sounds like the cats' lives are in danger, and even if they aren't killed, they are also not being treated right as pets. Please try not to take offense. I understand that you're in a tough spot and you don't know what to do.
You could crate the dog for a lot longer than five minutes twice each day. If he is crate-trained, he will probably just sleep while in there. Lots of dogs grow to love their crates as a safe haven and quiet napping spot. Then the cats could be allowed indoors during those times.
Now, about the training:
Conditioning him to accept the cats is going to be a hands-on endeavor, not just a matter of him seeing them through the crate door until he gets used to them. If it were me, I would keep him on leash indoors at all times, and allow the cats free access to their home. You can have short training sessions whenever the cats are in the same room as you and the dog, where you use body-blocking (getting between him and the cats and capturing his attention) and redirecting (asking for behaviors like sit and rewarding or playing a game like tug, which redirects his attention to you).
You can also teach him the "Look!" command, which will be very useful with his reactivity to more than just the cats. Start by asking him to look at you when there are no other distractions, and then treat. Keep doing this, and when he's 100% start adding in other distractions and doing it in more stimulating places (the garden, on walks, in the car), and build up to longer amounts of time where he's focused on you before treating. You can then start doing it whenever he looks at the cats, and reward when he breaks his focus on them to look at you. This will also start to build positive associations with the cats -- whenever they're around, he gets treats for being calm! As he gets better with it, reward him with attention, games, and treats any time that he remains calm when in their presence. But keep him on-leash at all times while doing this!
When you are unable to have him leashed to you or your partner (for which you might find one of those belt-attachments useful so your hands are free), you should put him in the crate or in a closed room. For instance, if you need to take a shower, bring him in with you and close the door. That way, the cats are safe. Keep in mind that every time he is allowed to react poorly to them (barking, chasing, fixating) actually reinforces that bad behavior -- so you want to prevent it from occurring at all.
Incidentally, how much exercise is he getting? How much time is spent on walks each day, and other physical activity?