I would trust indigo's suggestion. Depending on wether the bridle had fancy stitching or not I'd go for an alcohol based dye of the right color but those can screw something up permanently if you are not careful (like my countertop)

"It's really quite amazing what a horse will do for you, if he only understands what you want. 
And it it's also quite amazing what a horse will do to you if he doesn't." ~ Bill Dorrance