Thank you for letting me know more about your four legged family members, I must say I love their names! I'm pleased to hear that Bear and Edgar get along so well. And the fact that Kyo was getting used to being around Bear in the past is a good sign. There are a number of steps that you can take to make Kyo's reintroduction to your new house be less stressful than than the previous time he was there. I am sorry to hear that he would't eat or drink, that must have been stressful on everyone. Ok lets work on how to help Kyo through this. Firstly there are a few tools that will make the transition much easier, you may already have some of them..I suggest that you get some Feliway it is a pheromone product that is designed for use in the environment whenever a cat may experience stress, moving to a new house is a classic example of such stress. You should be able to buy Feliway at a pets store or even online. It comes in various forms (diffuser, spray). The next tools that would be helpful are more for keeping Bear as calm as possible when you introduce Kyo. There is a canine version of Feliway called Adaptil, I would use this in the environment for Bear, it also comes in various forms (diffuser and collar are the most common). Next to help you encourage calm behaviour in Bear I would suggest you use a long leash (6-10 feet) to be used as a "drag line", if Bear is becoming too excited you can quickly pick this up and get Bear's attention. Then make sure you have some tasty training treats for Bear to reward calm behavior when around Kyo. That is the information on tools that will help, next I will explain the process...
The first thing to do is choose a "safe room" for Kyo. This should be a comfortable room that will be Kyo's room alone for at least a week. You can share this room with him but you should limit Edgar and Bear from having access to the room. I would use the Feliway in the room as this will help ease stress. You know Kyo best so try to maintain his regular schedule as much as possible in this room for the first week - normal feedings, normal litter box and litter and normal play or snuggle routine with you. When you see that Kyo is settling in a little probably a day or so you can let him out to investigate the rest of the house without the other pets being present...preferably they could stay in another room for 20-60 min. This allows Kyo to safely investigate and build confidence and smell Bear and Edgar in the new environment. Also use Feliway throughout the house as this will help both Kyo and Edgar. After his time investigating put Kyo back in his safe room then let him repeat these sessions over the next few days and extend his time out as long as he is comfortable. Do let him have access to his safe room so he can retreat there if he wants to. After a few days or a week of this routine you can start to introduce the others. I would start with whoever you think Kyo will do best with, it may Bear since it sounds like they had more contact previously. Make sure to allow Kyo to have access to his safe room so he can escape if he is nervous.
The introductions should be supervised at first, so you can help reassure Kyo and praise him with pets or soothing words for being calm when around the others. Cats will often size up other cats from a distance so don't worry if they don't interact right away. If by chance they fight or play rough your presence is helpful to give them a time out, you can break them up often by making a loud noise like loudly clapping your hands, or by using a spray bottle with water.
As for how to work on keeping Bear calm, it is absolutely normal for dogs especially puppies to want to play and be rough. What we want to try to avoid is Bear chasing Kyo because this chase is reinforcing and it will be hard to moderate Bear's energy. So we need to stop the chase before it happens. The way to do this is to supervise initial interactions and keep a drag line on Bear - basically this just means leave the leash attached to Bear's collar. Initially you may need to hold onto the leash but as Bear relaxes you can let it go and drag on the floor. The idea is if you see Bear is starting to become to exuberant you can quickly pick up the leash and redirect Bear's attention. This is where the dog training treats come in....they are reminding Bear that you are much more interesting than playing rough with Kyo. The most important part of the whole exercise in introducing Kyo is building his confidence - letting him know that he will be safe in a new environment with other pets. So to prepare Bear work on short training sessions with Bear, ask for a sit and reward with treats. Then when you have Kyo and Bear together Bear will be ready to respond to you asking for a sit or attention because Bear will know a treat is coming. The other way you can use treats is to reward calm behavior, once the pets all start to spend time together when you notice Bear is being calm (perhaps laying down, chewing a toy or sitting) you can reward with a treat, this will remind Bear that chasing cats is not as fun as being calm. It is a good idea to keep the treats handy.
This process will take a few steps, but the more time you can dedicate to making Kyo feel safe in the beginning the sooner everybody will be happy in the house. For at least a month I would not allow all pets to be together when you are asleep or out of the house. At these times I would put Kyo in his safe room. Doing this is for everyones safety and it will hopefully avoid Kyo experiencing any scary interactions that could make the introduction more difficult. It is great that you are looking for ways to help Kyo through this time, change can be scary for cats but taking your time will help avoid this stress causing Kyo any health problems. One thing to make especially sure to do is to provide one litter box per cat plus one (3 for your household) the reason is that cats are particularly fastidious when it comes to litter and not using going to the bathroom can sometimes cause bladder damage. But giving Kyo and Edgar 3 boxes (preferably with clumping not scented litter) that are cleaned daily will greatly decrease any stress.
Please let me know if you have any questions :)