Not 100% sure if this is what you're after.
When I use an external intervalometer connected to the bulb port, when I just want an external shutter button to avoid vibration (but no bulb setting) I just set it to manual mode (or which ever mode you prefer to use) and dial the exposure settings I want on the back of the camera.
If you do want to use bulb, I guess the minimum exposure is how little you press down the button.