I wouldn't want it darker. Maybe shinier. I am not a wood working guy, but I thought it said "stained and lacquered". I can't find where I read that at the moment. But could I add something like clear lacquer or clear coat (no clue) to match the sheen of the H3's?
How shiny? If you decide to try this, you could buy a small can and test some polishing methods, to avoid damaging the speakers. Or, you could ask if they would send a piece of scrap wood with the same finish, so you can practice. Altering the finish will likely void the warranty (if it covers the finish), but it could make the center match the Klipsch better.
FYI- sandpaper is available in grit beyond 12,000 and that's fine enough to polish bare wood.
Edit-
I used the online chat on the Crites site and asked what they use for the finish- Minwax stain and polyurethane- they provided photos of the cans but they don't use all of them at one time, but they use a combination of the stain and poly.
I have polished polyurethane and have used mostly Minwax- I sprayed my kitchen cabinets with their Fast Drying Satin Floor Polyurethane and left it alone, but I can polish an inconspicuous spot and let you know how it turns out.
Poly is different from most other finishes WRT adding coats- it needs to be re-coated within a certain amount of time (several hours), or after 24 hours and if the latter, it must be sanded (or you can use ScotchBrite pads).