Mice coming through gas fireplace vent
Hardware cloth will work over the vent out, but if you believe they are coming also (or rather) from the hole the vent makes in the outside surface of the house, you can use a copper sock to fill that hole around the vent pipe. For a small job like a vent, the copper sock can just be one of those dish cleaner copper pads (they unroll to about 3' long). The copper will not rust.
I dont think that steel wool will burn at the vent exit, but it will rust. If the steel wool can burn, why wont the house catch fire with or without the steel wool?
Why are they coming in the vent? Is it close to the ground? Is there landscape plants near that vent? Is the outside siding, or brick? Does the vent have a flap?