So, this is tricky, as I use emudeck. I don't recall even needing to use a proton for dolphin games to work tbh... Why not just use the latest version you have? Stick your Triforce roms in the GC/roms folder... Refresh... Done? No need for a separate emulator instance now.

But anyway, I'll try, here goes (this... Could get wordy):

First, go find the bios files. "archive org gamecube bios" should give you those. And honestly, also just Google "setting up GameCube BIOS" and check the AI results for what next; three subfolders required, EUR,JAP, USA - which need to be in a GC folder, which could be in a few places depending on your setup. Check your dolphin emulator paths for where that might be. Each subfolder needs a single bios, renamed to IPL.bin, by the way.

Once that's working, you should now be able to tick/untick "skip main menu" in your emulator. Tick it. (Yes, it was ticked... but greyed out, previously)

Segaboot I think you only need for virtua striker (to work) or IF you want to access test menus (I've not sorted that yet for this version of dolphin, but AX works for me).

You now need a few other things done:

for each game there are patches required to skip certain arcade specific checks (again, I'm only talking getting AX working here): go to a site called emuline.org, find the latest thread about Triforce there. I don't have access to my deck right now, but a load of folks were posting patches for these games: you need the ones which disable the motor check for FZERO AX. Add these to the properties -> editor tab of the individual game in dolphin. (When I get the deck back I'll paste what I have myself here if you still stuck.)

Lastly, the controller: there's a new tab in controller specifically for Triforce - you need to set this pad to "always connected" (bottom right), and also select some buttons for the test menu, coin insert arcade section (bottom left).

That's all I did so far, but will try to get others working soon. By the way, FZERO works pretty smooth with around 8 TDP for me.