My best advice is ditch the gamecube and go with a Wii. You are so early into the project that it won't be difficult to change over. The information in guides on Modretro is very, very outdated. GameCube portables are obsolete, and the only people making them now are only doing so because they are not properly informed. 
Here are a few of the reasons why everybody makes wii portables instead
-WIIS DO NOT NEED REMOTES TO BOOT GAMES.
 PortablizeMii makes that easy.
-Wiis have essentially 
perfect compatibility with gamecube games
-Wiis can be trimmed 
smaller with less effort
-Wii portables
 cost less to make
-Wii portables are 
much easier to wire up
-Wii portables do not require you to wire a wiikey fusion or a memory card
-Emulators run better on wii
-Wiis play Wii games, virtual console, wiiware, emulators, and homebrew
-Wiis use 
MUCH less power, so the 
battery lasts longer
-Wiis create 
less heat, making cooling them easier with smaller fans/heatsinks
-Wiis use usb drives, and are not limited to 32gb sd cards like gamecube portables
-Wiis have 
more documentation than gamecubes for portablizing
There are many more reasons to build a wii. The only 2 reasons to build a gamecube portable are bing misinformed, or "because I want to." Yes, GameCubes have the BBA adapter and GameBoy player, but the utility of those in a portable is negligible.
Here's a guide to get you started