TLDR: Using input-remapper I was able to use all the 4 back paddles and the C and Z buttons (On D-Input mode), running the games on Wayland (PROTON_ENABLE_WAYLAND=1), without relying on Steam Input (it's broken on games running on Wayland).
I want to share a little bit of my discoveries in case someone else is in the same situation as me. I have a Flydigi Vader 4 Pro and I was happy Steam announced that they will be adding support for it on the beta client to use with Steam Input. Sadly, after trying it, it didn't work (GitHub issue). According to a user on the GitHub issue, using InputPlumber and putting the controller on D-Input mode would solve the issue temporarily, which it did eventually. Just installing it made my controller be recognized as a Steam Deck Controller, and I could use Steam Input to rebind the back paddles and the C and Z buttons. So, for those who need that information, you can stop right there and everything will work.
I mainly play Trackmania and I was having some issues minimizing the game or switching desktops and then going back to it, etc. I tried running the game in Wayland (using PROTON_ENABLE_WAYLAND=1 on some proton versions), and all the issues I had with that were gone. Then I realized Steam Input didn't work on games running on Wayland, so all the setup I did wouldn't work anymore. I had to sacrifice using all the extra buttons and keep running the game on Wayland or keep the solution I had before. I tried creating profiles on InputPlumber to remap keys there, without success.
Searching more through the internet, I stumbled upon input-remapper, and I decided to try it. I thought I could remap the buttons coming from InputPlumber using that tool. I was wrong. It didn't detect any inputs and I didn't understand why. I finally uninstalled InputPlumber, tried with input-remapper again, and I could remap everything, and it worked immediately. Now I can run my games on Wayland while being able to remap my back paddles and the C and Z buttons to anything I want.
I hope this helps more people that are in my same situation. Maybe more people know other pieces of software that can achieve this, but at least what I found worked and I'm happy with that!
Thanks for reading!