Hmmmm, I think that this time I can give a you advice, in exchange for all the help you've given me Kelley, myself being a 2 bird owner. Although my case is extremely different since my birds aren't trained, I certainly do have some experience with 2 birdies behavior and introductions since I got my 2nd bird after a week of owning the first. From the same store so I didn't do quarantine (hopefully nothing bad happened).
I think you should try out a few things before taking any action (I mean moving cages together or birds together). Firstly, I think you should DO take Levi to meet the new, bird. Take Levi out of his cage. Then take him into the room Ziggy is. Make sure that Levi is WITH YOU and not just flying freely, as he should understand that YOU are introducing him to another member of the HUMAN flock (if he thinks he's a person), although this HUMAN looks more like him hehehe. You should let this introduction/interaction last for around 15 min or more depending on Levi's attention to the other bird. Be creative, take Levi's toys so he can play with them when the two birds are watching each other, Like you place Levi on top of Ziggy's cage and he plays with his ball or is eating some millet, or is taking a bath WITH YOUR aid.
After watching their interaction then you should give observations to BOTH of your birds reaction. Do this introducing several times, not just once, so you can conclude what is really happening (Levi not caring, or Ziggy just getting all excited, or both birds getting excited etc.). The next step should be, move one of the cages into the other cage room, I would choose the room they will be on after they live together to be the one you move the cage into. If you're planning on Levi's room take Ziggy's cage into that room. However this decision should be based on what were your conclusions in the previous observation.
That is for example, If everytime you took Levi, Both birds showed interest, but Ziggy was always excited to see Levi, and after YOU took Levi back to his cage how did Ziggy reacted, wanted to go with him and WITH YOU, or just stayed calmly in his cage after the fifth time Levi went back because he got used to this happening, etc.
So if Ziggy always wanted to go back to the other room with you guys, then you can definitely move his cage into Ziggy's room.
After you have move the cage to the other room. Again observe which things change, Does Levi goes more to Ziggy since he is in his room or since he knows him better etc. At this step (with both birds in same room), it is crucial that you notice if there is any change in Levi's behavior, is he more birdy like or still thinks he's human. Because here's your chance to watch Levi stepping up to your hand, watching how YOU and LEVI trust each other and will see a feathered friend trusting YOU so hopefully he'll learn to trust you a bit more. Continue your training as usual with Ziggy, especially if you have Levi out, Show Ziggy that to get to play along with Levi he should learn to trust you. Watch out with that cage door though, make sure to cover it with your body or something so he doesn't try to just sneak past you when you're training him.
At this phase, keep on observing. Does Ziggy is more crazed out than before for him wanting to be with Levi, or now that they're in the same room he's more calm? Be with the cages in the same room and separated birds as long as you need to. Hopefully at this stage it will be easy to train Ziggy. So when you train him you can give them both out time together and see how are they getting along.
Now after this it's entirely your call. If you see your birds are getting bond quickly then let them be and show Ziggy the new cage etc. I think he'll naturally will begin to move in with Levi. I don't think there's much risk of Levi going back to the old cage since he has his bigger cage and such. Also if you have trained Ziggy out, don't have any toys near the old cage. This is just some ideas that come to me that you can try before trying to move the birdies together.
I think that hand training Ziggy will prove difficult, so maybe you will have the 2 cages together for some time, always be patient.
Keep in mind that you must show Ziggy how Levi trust YOU and your kind. Keep an eye out for any "disobedience" by Levi after he's been interacting with Ziggy, and take him back to his cage for a training review
It is expected that Levi will be giving Ziggy more attention each time, just be sure that he's always carrying out your orders. He steps up every time etc.
Have a keen eye with observing, you will know what to do when the time is right (move the cage, etc), you know your birdies
Also when taking Ziggy into the new room cover 3 sides and top as you already know, for a day it should be fine since the rooms are connected and it isn't a complete mystery to him.
I hope this helps, if anything else comes to my mind I will let you know.
P.S. Also if you see they're interacting greatly when you take Levi to Ziggy's cage maybe try putting Levi together with Ziggy for a short time. And do make sure he listens to you when you want to take him (Levi) out.