The only difference between MIDI_SYSTEM_DEFAULT and MIDI_SYSTEM_GM1 currently is that MIDI_SYSTEM_GM1 disables several events/controllers that weren't included in GM1: MIDI_EVENT_RESONANCE, MIDI_EVENT_RELEASE, MIDI_EVENT_ATTACK, MIDI_EVENT_DECAY, MIDI_EVENT_CUTOFF, MIDI_EVENT_BANK, MIDI_EVENT_BANK_LSB, MIDI_EVENT_VIBRATO_RATE, MIDI_EVENT_VIBRATO_DEPTH, MIDI_EVENT_VIBRATO_DELAY.
If you would like to keep those events disabled but still use a different bank from a soundfont then you could use the BASS_MIDI_FONTEX(2) "sbank" member to set which bank in your BASS_MIDI_StreamSetFonts call.