hehe, well, my guide is pretty basic.
thank you for the credits anyway
well, the batteries are one of the easier parts, I am surprised you already made the hard parts like controller cut and wiring and stuff like that with no issues but had trouble with the batteries
well, I recomend,to use this:viewtopic.php?f=41&t=32713
if you have a rev A gamecube, you will need to wire the batteries directly to the gamecube power/regulator board.
about the screen, if it works on 12v , you can also connect it directly to the screen.
then add a switch at the + (red wire of the battery pack) wich will work as the main switch to connect the batteries to the whole system (gamecube+screen)
also make a connector to charge the batteries with an external charger (I highly recomend to not play meanwhile recharging batteries)
14.8v battery packs works pretty nice for gamecube laptops.
if you want something simple, you can use this:
battery (4400 mAh may got your gamecube working arround 1-2 hours on batteries, if you want larger play time get 2x those batteries and wire the packs in parallel)http://www.all-battery.com/li-ion186501 ... ector.aspx
you can charge them with this chargerhttp://www.all-battery.com/universalfas ... cells.aspx
you can also get this:http://www.all-battery.com/conversionca ... ector.aspx
the wiring is pretty simple, I'm sure you will get it with no issues
just check the polarity twice before connecting the stuff and everything will be OK ^_^