Agreed, you should ALWAYS start out with 1 cage for every 1 bird you have. If you are an experienced bird-keeper as you say, then you will already know this, and know that you should never, ever, ever bring home a new bird and just immediately put them into the cage with any bird you already have. And this is for a number of reasons; first of all, you have to keep any new bird in a separate cage alone in a totally different room for a 30 day quarantine to protect your first bird from any possible infections and disease...
However, as Triggs and Christa have already said, there is absolutely no way to guess how two birds are going to react to each other, no matter the species or the gender. So no one can really answer the question you're asking with much certainty at all, because no one is able to guess how each bird is going to react to the other. They may hate each other and become immediately aggressive and violent. They may hate each other but simply tolerate each other. They may be totally indifferent to each other. They may like each other, but not bond closely. They may bond very closely, but not in a way where they want to mate (if an opposite sex, same species pair). And they might bond closely and want to mate. But there's absolutely no way to anticipate this.
If you already have a male that was a former breeder, then the first question is going to be "What is your goal?" I'm assuming you don't want to breed him since you're asking about what sex-pair makes the best pair. Basically the bottom-line is that you need to approach this with the assumption that you will not be able to house the new bird with your current bird, and that they will each be inside of their own cage. As far as their level of interaction with each other, there's no way to anticipate that before you bring home the second bird. but as far as what sex is "most likely to get along with your male", again, there's no way to know. And if you add a female to your male's cage they may bond closely and start mating and producing chicks, which is something you definitely don't want to do if you have no bird-breeding, hand-raising, hand-feeding experience/equipment, etc.
Never buy a second bird for your first bird. This is a really good rule to follow, as you should only ever get another bird because YOU want to get a seond bird. And in that situation it shouldn't matter whether or not the new bird gets along with your first bird. It can't matter, because you can't ensure that it will happen. So if you do go and get a second bird, have a separate, adequate cage set-up and ready for it, in a room separate from your current bird, with the cage having it's own food and water bowls, lots of toys, foraging activities, etc. And keep the new bird in that separate room for 30 days, monitoring it for any signs of infection or disease (A wellness-exam should be done by an Avian Specialist Vet within the first 30 days anyway). Then after the 30-day quarantine is over, only then can you introduce the new bird to your first bird, but only by moving the new bird's cage into the same room that your current bird is in (or move both of them to wherever you want to keep them). Put their cages in a spot where they can see each other well and talk to each other, and then see how their interaction with each other goes. If they are talking to each other and are interested in each other, then you can move their cages right next to each other, and see if they start spending time sitting close to each other inside of their respetive cages. At this point it might be obvious that they really like each other, and it may be obvious that they absolutely hate each other...or, they may not pay any attention to each other at all. After they are gradually introduced and have spent some time together in the same room in their own, separate cages, then you can try a supervised out-of-cage-time, and watch them very, very carefully to see how they interact with each other.
I know you mentioned that you have "animal experience" and that you "know how to introduce them to each other", however, if the only bird experience you have is owning your current male Kakariki and that's it, if you've never introduced two birds to each other before, then you need to know that it's completely different than trying to "introduce" other animals. Birds need to be thought of in-regard to their interaction with each other like humans do. They are extremely intelligent and sensitive, and they like who they like, and they dislike who they like, and everything in-between. They are very highly-developed when it comes to relationships, emotions, etc. So the advice that Triggs gave to you was exactly the correct way to answer your question; your question has no definite answer. Each bird is an individual, and because you have no way at all to anticipate or guess how they are going to react to each other, you absolutely cannot bring a bird home and put it into your current bird's cage. Just doesn't work that way.
