If your bit has aurigan rings then use a bit of diamond paste on the rings if you want to shiny them up. Never use a cleaner on the mouthpiece, just wash with water (you can't see the tarnish on mouthpiece when the bit is being worn anyway).
I thought the tarnish was what was supposed to taste nice? I just wash mine in water. If she has been eating with the bit in, I may use water with washing up liquid then rinse in plain water.
But then, I buy the ones with stainless bit rings, so the outside looks good.
Mine get rinsed with water after use and when I take the bridle a part for cleaning, the bit us cleaned with washing up liquid. I have used a specialist paste but found washing up liquid was fine.