Before I get anyone's hopes up, I should state that this did not work. If that disappointment isn't too much for you, I'd love it if you kept reading.
So now with that out of the way, I'm sure you're wondering, what did I do differently this time?
1. A different method of obtaining the system files
The first major change I made, which fixed the biggest issue from last time, was the way I obtained the .app files within the Wii System Menu install WAD. Last time, I chose to just save the decrypted contents in NUS Downloader, however that ended up preventing me from packing the WAD at the end of the process. This time I didn't save the decrypted contents, but instead had NUS Downloader pack the WAD for me. Once I had the WAD, I unpacked it with Sharpii, which left me with almost the same files. The difference is that this time I had the ticket[Check] which allowed me to pack the WAD at the end.
2. Recompressing the archives
Note: Since this article was written long after I tried the things listed in it, accuracy is not guaranteed.
What I tried this time was enabling LZ77 compression on the .arc file, then compressing the .arc again into an .ash, rather than going directly to a .ash. Thanks to my new method of obtaining the Wii Menu files, I was able to pack the WAD no problem and install it. However, once I installed the WAD in Dolphin, problems arose. The initial setup went just fine, as it should have since I didn't mess with those files, however once I tried to select the Disc Channel, it all came crashing down, in a way more literal than I would have liked. Fatal exception. Unfortunately, my .ash recompression method had not worked properly. (For the sake of making sure the WAD itself wasn't the issue, I unpacked, repacked, and installed an unmodified version and it worked just fine.)
So where does this leave me?
When I originally started (and then lost) this article back when I did all this, it left me with no leads and nothing else to try, or at least nothing I could think of. Thankfully, a fellow nerd known as Garhoogin gave me some extremely valuable information. It turns out I wasn't understanding the Wii's file formats correctly (specfically the .arc and .ash formats). Only .ash files are compressed, whereas .arc files are just archives. That's why adding the LZ77 compression to the .arc didn't help; it's not a compressed file. So now my new plan is as follows:
So, for now I leave you here. Hopefully there will be an update soon, as I intend to follow the plan above in the next few days so that I can make an update to this faster than last time.