Thanks for posting. Just to make sure we're talking about the same part of your fireplace, the "firebox" is the area in which the fire would be burning, right?
Fireboxes are very specifically designed & built to keep fire from escaping the firebox and to control the tempurature of the exterior surfaces of the firebox that may come in contact with combustibles. Second many types of stone, including slate, can literally explode when exposed to high heat. Moisture trapped within the stone has no where to go. Absolutely! Positively! Do not alter your firebox or redesign your fireplace!
You want to err on the side of caution here! Put a beautiful glass door on your fireplace, dress up the hearth, mantel, and surround of your new fireplace as much as you want. Slate would be beautiful on the hearth and/or as a surround so use your imagination on these surfaces that are not affected by fire.
Hope this helps!