Is there a doctrinal error in objecting to compelling a fearful Muslim worker to slaughter, when there is an alternative and out of respect for the feelings of a Buddhist worker? And is it a condition for slaughtering pigeons to feed falcons that the conditions of slaughter, such as facing the Qibla and pronouncing the Name of Allah, be met?

It is permissible to feed carrion to falcons, birds of prey, and any animal that is not eaten. So, pigeons that have died can be given to falcons. It is not permissible to kill an edible animal, such as a pigeon, except by lawful slaughter (dhakāt shar‘īyah), whether for human consumption or to feed birds of prey. If an edible animal is killed without lawful slaughter, it is considered carrion. It is a condition that the one who slaughters be a Muslim or a person of the Book. Your brother is not allowed to force the worker to slaughter unless there was an agreement with him to that effect.