Remember you have a duty to keep the cats safe. If you are only going to have the cats for a short time, there is no real point in intermixing them. Cats can be happy in cages, tho' dogs can't. However, if this is to be long term, then they will eventually have to meet. The sooner this happens, the better. It serves no purpose waiting because animals don't have any sense of time passing, but can become stressed knowing the other animals are behind closed doors.
It would be best to introduce the docile dog to one cat and then the other.
Create a "safe zone" where the cat can run to escape the dog, if needed, such as a high place the dog can't reach, etc. Make sure the dog is leashed and calm and sitting or lying down (best if he is asleep). then, simply open the door of the room or cage the cats are in and let the cats decide if they want to join you. Don't hold or touch the cats, pretty much act as if you are ignoring them. However, keep the dog on a leash and correct it by touch without using your voice, if it becomes excited about the cats. Also, if you can't calm the dog down again with the cats there, have someone remove the cats. Repeat this a few times a day until they get used to each other. Don't rush it.
The dogs and cats will know that the dogs are the ones who "own" the place and the cats are the interlopers, so when you do allow them to come together, stay with the dogs and treat the cats as "strangers". Let the dog know he can't chase, harm or bark at the cat.
If this goes well, introduce the other dog to the cat(s). Until you and the cats trust the dogs, make sure you chaperon all encounters between them.