You'll need to get a copy of ESR, place it on the memory card, and point FMCB to it in the configurator. Then, FMCB should recognize when the inserted disc is ESR-patched and will launch ESR, which will then boot the game. For this purpose, I recommend the non-GUI version of ESR r9b. This way, the ESR splash screen will come up for a few seconds and the game will boot with no input required. Also, if you're not aware, it is impossible to boot unpatched backup games on the PS2 without swap tricks or modchips. They must be ESR-patched, and they must be DVD+/-R media. CD games would have to be converted to DVD and then ESR-patched.
Chip-free/MC-free homebrew <3
FMCB 1.93 on MC for emergencies, NA
250GB HDD/FHDB 1.93/OPL 0.9.2 (GSM and non-GSM)/PS2 games, NO HDDOSD
80GB HDD/FHDB 1.93/POPStarter rev12/OPL644/GSM0.37/PS1 games, NO HDDOSD
SCPH-75001 (sitting idle, slims suck):
FMCB 1.92 on MC