Thanks to everyone for the tips. I did this a couple weekends ago and it worked perfectly.
The hardware store near my house didn't carry Minwax, so I used Olympic brand solid exterior latex stain. I ended up buying one quart of 'Copper Henna' and one quart of 'New Pilgrim Red.' Equal parts of each gave me the color I was looking for.
One caveat – Whistle8mp is right about being gentle. My belt got a little frayed just from me working the stain into the belt with my fingers.
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