I haven't worried ever about watch, jewellery, coins or keys when calibrating and have never had an issue. The compass is only effected by ferrous metals. Many mechanical compasses have stainless steel or brass enclosures. Your watch may have some ferrous metal however of its mechanical will be mainly brass and nickel in the movement with precious metals and/or stainless steel in the case/band. Coins and keys are also non ferrous.
It's easy to check the influence of a particular thing, fire up the bird and AC and open the sensor screen and watch the mod value as you bring it close to the compass. No change means it can't be having an influence.