Welcome to bitbuilt! I'll try to answer some of your questions
As for the Wii U screen, i can't advise on that but I recommend going through the worklogs and reading them as there have been users who have used the Wii U gamepad in their portables, so you should be able to find information there!
As for your power solution, there really isn't an inexpensive way to power your trim and it is more work to use the wiis regulators (if even possible) rather than buying one of the many great solutions people have out there. I really recommend saving up and going with the RVL-PMS 2 with the PMS-PD 3 from 4layertech.com as those will be your best options in terms of safety and features when it comes to powering your trim. The PMS-Lite is a little cheaper but doesn't offer as many features as the PMS 2.
There is a user on here who has an aliexpress store that sells a cheaper power solution but I cannot advise on how it works or it's features. Here's the link though if you want to give it a look:
As for syncing your wiimote you will have to relocate the Bluetooth module to be able to use a wiimote in your project. Also syncing is not done via software. If you have a wiimote already synced to your BT module, it will work if you relocate the bt module after you trim, however if you need to resync, or sync a new wiimote, you'll need to wire a button or something similar between 3v3 and the sync pad on the Bluetooth module. Take a look at the wii trimming guide and you'll see what I'm saying.
As for the gamecube controller: again, I recommend using the GC+ 2.0 from 4layertech.com as that will be your best option. Alternatively I believe the aliexpress store I linked above does have an RP2040 based controller solution, but I cannot advise on it.
And for a name, the running meme around here is using something with the 2 i's from the wii name such as Kill Mii for example, or i have a worklog currently with the name "Guitar Hiiro" so you can do whatever you want but that's the theme around here lol
Overall if your new to the forums, I really recommend reading and going over every resource you can on here as there is A LOT of information on everything. It's not to say you can't ask questions on specific things, but the creators and users of bitbuilt have made the process as streamlined as they can for beginners, so definitely take hours and days if you need to, to read on information provided.
I hope this helps and good luck with your portable!