I've done it before with older strap goods (bridle parts, stirrup leathers, lead shanks) before and had good results. I wouldn't do it if your dressage bridle has the white padding (it may discolor), or if it's brown and you don't want it to darken. It shouldn't cause problems if it's solid black or brown leather. If you have one of the embellished crystal browbands, personally, I would just wipe the leather parts of it well with oil, but not soak the browband. I disassemble the whole bridle, clean everything well, and soak it overnight. In the morning, I take a towel and wipe all the excess oil off. You may have to wipe it before use the next couple times you use it if the oil really saturated it. The hardware will NOT rust. Oil is hydrophobic and excludes water from whatever it coats; water causes rust but oil prevents it.