In case you still not able to got RE2 working, here what I've done...
First you'll need a Folder for Disc1 and another for Disc2
Disc1: Games/#/GameData/SLUS-00748.bin
SLUS-00748.cue
SLUS-00748.png
pcsx.cfg
Game.ini
Game.ini file must contain the next:
[Game]
Discs=SLUS-00748
Title=Resident Evil 2 Leon
Publisher=Capcom
Players=1
Year=1998
Disc2: Games/#/GameData/SLUS-00756.bin
SLUS-00756.cue
SLUS-00756.png
pcsx.cfg
Game.ini
Game.ini file must contain the next:
[Game]
Discs=SLUS-00756
Title=Resident Evil 2 Claire
Publisher=Capcom
Players=1
Year=1998
As I said before, you'll need to merge Track1.bin & Track2.bin on both Discs to get one .bin file per Disc (SLUS-00748.bin for Disc1 & SLUS-00756.bin for Disc2) in order to work properly.
Tuto how to merge .bin file
Doing this it will work every normally, you'll need to select from PSClassic Menu which Disc to play instead of selecting RE2 and "changing" discs
EDIT: I'll let you use the cover I'm using for both discs
Just change the name as I marked before and include it to every game folder...
SLUS-00748.png for Disc1
SLUS-00756.png for Disc2