Ok I have some great news! With my newest regulator for a WiiKey Fusion or WODE. The gamecube will draw a total of 1.7A (a bit lower in game) on 7.4v. This is the lowest I can get it with everything working 100%.

Let me explain:

The GPU takes 1.8v originally. I was able to power it of 1.7v while still working correctly. Drawing 3A at 1.7v
The CPU takes 1.55v originally. I was able to power it of 1.5v while still working correctly. Drawing 2.2A at 1.5v
The 3.3v line will stay the same as putting a lower voltage in won't make any difference in amp draw. Drawing 0.8A at 3.3v

By using a PTR08060 or PTR08100 for the 1.7v line
a PTH08080 or PTH08000 for the 1.5v line
and another PTH08080 or PTH08000 for the 3.3v line
we now have the possibility to bring down the total amp draw!!!

PTR08060 and PTR08100 need 100uF caps on output and input. PTH08080 and PTH08000 only need 100uf caps on input.

Following my diagram below you will be able to see which resistors you need to get the appropriate voltage.

The other regulators on this diagram are for powering the cube of a single 3.7v battery. Amp draw is not as effecient as 7.4v so not really worth your time.

The total draw of running the cube of these 3 regulators is 1.7A (a bit lower in game). If you don't want to bother with 2 separate regulators for the CPU and GPU. You can use the PTR08060 or PTR08100 on 1.7v to power both chips. The difference in total draw is 0.2A. As running it of those 2 regulators will give you a total of 1.9A on 7.4v

It's your choice. Note that the draw of the WODE and WiiKey chips have to be added. I still need to measure those.

My results confirm yours.

At 7.4v :

1.9v line using PTR08100
3.3v line using PTH08000
5v line using PTH08000
12v line at 7.4v

Total amp draw in game and booting up variates between 1.9A and 2.1A

This was with the WiiKey fusion running but no sound.

And my PTH08100 works fine with no 100uF cap on the output.

So is there any reason why the OKI-78SR series would not work for some of the power requirements? They can regulate 7V~36V down to either 3.3V or 5V, depending on the model, at 1.5A (based off its datasheet, Mouser has it wrong, but it's also 50 cents cheaper there than at digikey ($4.52 vs 3.93)).

Granted I still would need to use the PTR08100 for the 1.9, and figure something out for the 12V line (probably just run it off of the battery), but the entire power supply would run around $30, all said and done then...