There isno way to match the stone but a nice piece of soapstone maybe.But I have no idea why the hearth needs to be bigger if the insert is gas where I llive you don't even need a hearth the size you have for a gas insert. It is the mantel that will cause some issues since you will have gap when you move it so maybe the same soapstone to kind of make it look on purpose.

I would get some good pictures and visit a local stone yard. What is avaiable in anyones area will be different from what is available in anothers area. Chances of matching it completely (even if you knew exactly what it was) are slim to none as it is a naturally occurring product.

The fireplace is a focal point. The stone and the grout/mortar added will not match perfectly and may be very noticeable given its location.

Will your budget cover a complete redo? Also consider widening the drywall box over the mantel and drywalling the room ceiling.

Even if you found a matching stone veneer, the run of the veneer will never match what you currently have.

Check out local stoneyards; yours may be common and available. I like Patricia's idea to make contrast your friend.