Welcome to horseback archery!
The Centre of Horseback Combat have indeed left Sussex. Watch this space though, because there may well be possibilities in other areas opening up soon, including the Forces Equine Games, in aid of injured servicemen.
As for getting your horses involved, there are various people from the BHAA who would, I am sure, love to help. We have a couple of members in Somerset and more in Bristol. My wife and I are in Sussex and would be happy to come and lend a hand.
One of our members runs a riding school and he teaches natural horsemanship. He wrote an article for the Parthian Shot, our newsletter, on training your horse. I shall try to find a link to it.
For my part, I would say that the important thing is to do as you have been doing: get the horse used to bows. Shoot near them, wave bows and arrows around near them, including when you are up on the horse (you'd be amazed how many horses, even good archery horses, spook when somebody hands arrows to the rider). Also ensure that the horse is used to being controlled with the legs only and will not go crazy when you let go and start shooting. Jumping at the arrows, following the arrows and just plain taking off when the pressure goes from the reins are all potential problems.
Depending on who is going to be riding and what events you plan to do, it may also be worth trying to ensure that the horse is to some extent desensitised to minor leg control and twisting whilst being shot off. Most people will not be able to maintain beautiful control whilst shooting, especially when shooting backwards or upwards.
When it comes to shooting, the bog thing for mounted archery competition is nocking speed. You must be able to draw an arrow from you quiver (or bow hand or draw hand, depending on style and event) and get it onto the string without looking at it. You will see from the video that Kassai holds his arrows in his bow hand. He's terrifying good at it but bear in mind that whilst it is, when done perfectly, very fast, it is also not allowed in some competition styles and Kassai does less well in those events (don't get me wrong, I've shot against him and he is awfully good from a quiver as well, but others are better, which you can't say about his style). Kassai's method is also less safe than a quiver. I would always recommend a quiver to start with.
Personally, I shoot with a thumb draw. Kassai uses a Mediterranean release and he is emulated by most Europeans. We have members using both, as well as the Sassanid style, which is kind of a hybrid between the two. It's worth trying them all. If you use a Kaya then I would definitely recommend trying the thumb draw - Mediterranean is generally reserved for longer bows.
Hope this late-night ramble helps. Do let me know if you have any questions.
Dan (BHAA Chairman)