You should never see an "insert disc 2" screen. Make sure you're following the instructions. Get to the very end of disc 1, save, and then load that save from disc 2 with the code enabled. Disc 2 should have been created with the same Disc ID as Disc 1 so that the save can be loaded.
MGS Disc Swap Fix Version 2 by psychospacefish.
Sorry it took so long, it took way longer than I thought to hack this one.
You will need the latest CWCheat.
1.)Play MGS Disc 1 until you go through the Lvl.6 door in the Snowfield. Don't go down the stairs, save your game, then quit. Put the Disc 2 EBOOT on your memory stick if you haven't already.
2.)Open ms0:\seplugins\cwcheat\cheatpops.db in notepad. Add these lines to it
_S slus_00776
_G Metal Gear Solid Disc 2
_C0 Disc Swap Fix A
_L 800B7518 0002
_C0 Disc Swap Fix B
_L 800B7520 0002
3.)Start Disc 2 and enable ONE cheat, load your Disc 1 save. If it doesn't work try the other. Now run back into the Comm. Tower, and back out into the Snowfield, then continue down the stairs into Disc 2.
Note: After you've used the cheat to load your game, you don't need to use it again. This will load ANY Disc 1 save on Disc 2. Don't use it earlier than you need to or the game will freeze when trying to load a cinematic.