Yes, it is possible. You need the
Smash Stack exploit (copy the "private" folder straight to your SD card), the Gecko OS boot.elf (download
here, get the boot.elf from Gecko1931/Gecko1931/HBC/Gecko1931/boot.elf and put it in the SD card root), and a GCT file (put your codes in
here; the game ID is RSBE01 for US versions, RSBP01 for PAL, and RSBJ01 for Japanese). The GCT file then goes in sd/codes/. Your customisations, if you're using the file patch code, go in /private/wii/app/RSBx/pf/ (replace x with E for US, P for PAL, or J for Japan - the PAL code lets you switch between those three and /RSB4, but defaults to /RSBP).
Run the Smash Stack exploit (load Brawl and go to Custom Stages) and it'll boot into Gecko OS. Go to Config Options and switch SD Cheats to Yes. Save, press B, and go to Launch Game. It should boot into Brawl again, but now running your custom files and settings.