Write your TAS using BizHawk with the mGBA core

run this python script to convert the bk2 to a movie for Game Boy Interface (it will output as GBITimestamps.txt)
py -3 thescript.py yourmovie.bk2
https://pastebin.com/6jaPXm4H

Rename+move the GBI output to an SD card for playback and set it as the movie in GBI's CLI file
--movie=yourmovie.txt

Run GBI with whatever method you use to load GameCube homebrew

(optional) you may need to clear your cartridge's save data to defaults before attempting playback.  Separate cart dumping and flashing is not needed due to GBI.  GBI can backup the cart's ROM and save data, and can flash mGBA saves to console with the extension changed to .fla.  The menu for this can be accessed in GBI-SR by putting default.mb.gz in the GBI folder and adding default.mb.gz to the CLI file.  This can also be useful to dump the initial ROM before TASing a game to ensure that you're using the exact revision while TASing that will end up being needed on console

For assistance with setting up GBI:
https://www.gc-forever.com/wiki/index.php?title=Booting_Homebrew
Likely specifically using the SD Media Launcher method
https://www.gc-forever.com/wiki/index.php?title=Datel%27s_SD_Media_Launcher