You need to reevaluate the difficulty you believe this project is.
1. This step is by far the hardest and I would need someone else to help.
We are happy to help people troubleshoot and offer ideas and suggestions, but what you're asking for is a very time and effort intensive redesign of a complex circuit. While reverse engineering a circuit is generally easier than engineering one, something as complex as this would still be supidly difficult to pull off. Espescially if the joycons are four layer boards, which I suspect they are. Don't expect to receive a whole lot of help with that.
2. The board isn't from scratch, it would literally be a clone of the original joy con PCBs.
They'd need to be quite a bit different, since GameCube buttons and sticks have very different proportions from original joycons. Not to mention workaround for things like R3 and L3 due to the fact that GameCube joysticks don't have buttons built into them.
3. Solder wick makes connecting small components easy.
Solder wick is a useful tool, but totally useless when it comes to trying to o hand solder BGA. I'd recommend doing a good deal more research into soldering, since hand soldering BGA is probably one of the hardest soldering jobs to pull off, and you don't seem to have a whole lot of experience with soldering to begin with.
. This step just takes time.
Time, energy, probably a good deal of money since small mistakes are pretty much inevitable for just about anyone.
Well yea, most of it can be done pretty easily if you know what you are doing.
What I'm trying to get you to understand is that if you knew what you were doing, you'd see that this isn't a project that can be done pretty easily. Like Madmorda said, don't overcomplicate what you're trying to do. A much better option for a project like this would be to look into cheaper joycons, and build those into a GameCube shell. Yes, it wouldn't be nearly as streamlined, but it would be a good deal cheaper and less complicated.
Your overall idea isn't a bad one. It would be awesome to have PCBs that could slide right into GameCube halves for GameCube joycons. What I don't want to have happen though is have you go out and buy expensive joycons and equipment thinking that all you need is time to complete the project.