I strongly believe that Claybar doesn't work for the stains, as they are only good for removing a film of dirt/oxidation.
Polishing can work, but must use the right compound with a good level of abrasion - some brands call them light/heavy cutting compounds,even 'scratch removers' work. Bird dropping stains are more suited to a chemical + abrasive compound IMO and remember to polish outside the stain so you don't get an odd looking spot.
A good hand polish of about 2-3 mins will get rid of it if it's not serious - otherwise, it just takes longer. If doing by hand, I feel it's easier if the applicator is the terry-cloth weave type instead of a normal waxing sponge. Even a normal microfibre cloth works better than the sponge.
Shampoo-ing also doesn't help much because there is not enough work performed on the stain unless it's very fresh.
Edit: Before you run out to get any polishes, look through your arsenal of waxing products. Some may already have mild cleaning + polishing properties. No harm giving them a try one after another.
1) Wash the area to remove dirt
2) Get the mildest polishing product you have with a microfibre terry cloth/applicator.
3) Apply over the stain and polish with circular motion (with *moderate strength*), remembering to go around the surrounding areas
4) Rinse, check for stain
5) If stain persists, try a stronger product that you may already have, (go back to step 3 with a new applicator)
6) If it doesn't work, your product is not strong enough. Try something stronger.
