A hand held meter can be great for getting your exposure just right for your subject, whether it's flash or ambient light. The key is that is measures incident light (the light falling on the subject) not reflected light (the light reflecting off of the subject).
But many light meters can also be used to measure reflected light, good for measure the background. But your camera has a built-in 'reflected light meter' so you could just use that for the background.
In your example, you could aim the camera at the scene/background and adjust your settings until the meter reads -1.
Now you can use your flash meter to test your flash's output. Dial in the ISO that is on the camera (on the meter) and adjust the flash power or distance until the reading until the meter gives you the same aperture that the camera is set to. That should give you a properly exposed subject (from flash) and a background that is one stop underexposed.
But yes, if you need to adjust the ambient part of the exposure, then you can use the shutter speed because it won't affect the flash exposure.