Stories about persons of African, Asian, Middle Eastern staged in the past do not have to include encounters with ignorant or racist persons to be historically accurate. In fact the stories don't have to take place in Europe or "white" communities in America at all. Believe it or not, Black Britons did visit other areas--Darwin talks about meeting Black British sailors and one of his instructors, John Edmonstone, was a freed man of color who taught the skill of taxidermy.
Is it possible that real problem many fans have with the introduction of a Black Doctor is the fear of a proliferation of racially insensitive or ignorant White American/British characters like Baines, Joan-you read that in a book-Redfern, and Rocastle?
If so, what can be more racially insensitive than the assumption that because institutions like slavery, immigration discrimination, and Jim Crow existed in the past, all white people subscribed to societal racism?
European colonist protested the slave trade in their native countries and in the American colonies. In the 1400-1500s when Queen Mary and Elizabeth I were vilified by writers and critics outside of England, French, Dutch, German, and Portuguese authors and historians wrote about Angolan Queen Nzhinga and Princess Amina of Zaria (Yes, the books and 'histories' were reflections of what continues to be a European fascination with the so-called exotic, but Amina and Nzhinga were real people and rulers.)
Imperial European nations engaged in a love-hate relationship with the moral issue of slavery, freedom religion and culture, as well as simple appearance with nations they conquered. And most nations of the world still struggle with welcoming people from certain nations and faiths into their countries.
Still, At various times Africans and Asians in European society found freedom and mobility as a result of religious conversion or what their European "host" considered exemplary service or talent. Alexander Dumas stories were drawn from his Father's stories as a soldier in the French "Negro" Corps under the command of this Gentleman.
Elizabeth the First, after hearing of Hawkins and Drake's savage raids on African villages, predicted the Heavens would retaliate. Over half of the Abolitionist Groups in the USA were "white" Americans, and nearly all of the members of UK's Anti-Slave Trade Groups, Abolitionist groups, Anti-Caste Society, and Anti-Lynching group in the UK were white. Although a scary large number of persons possibly rose as one with Enoch's River of Blood speech in 1968, according to reports an equally large number of persons were appalled and stood against it.
Mary Shelley, whose parents were actively involved in the British Anti-Slave movement is said to have drawn inspiration for Frankenstein from the deplorable conditions many newly freed British Slaves a vindictive sugar and shipping industry subjected them to after the Anti-slave trade law was passed.
African and Black British businessmen, poets, musicians can be traced to Elizabethan time and before in the courts of Elizabeth and her Uncle by marriage, James of Scotland who seems to have given a generous sum money to a mysterious woman of African descent. Elizabeth I's letters and financial statements indicate the Queen was fond of and heaped gifts of clothing and (possibly relative trinket not royal) jewels upon her personal "African" maidservant/musician.
Queen Victoria wrote with true affection for her adopted African Princess ward and her God-daughter Victoria Davies. She welcomed abolitionist and writer Josiah Henson, and expressed her support in “his peoples” fight for social equality. (Yeah, this is the same Queen Victoria who ruled over a government that held much of the African and Asian nations in colonial bondage-go figure.)
The most loved stories of persons of African, Asian, Polynesian, Aboriginal, Maori, South & North American Indigenous descent--told by Persons of Color or these Nations --are not about the power of oppressive societies but Humanity's tales: An individual's triumph over adversity, struggle to achieve ethical courage, devotion to family, romance and love.
Perhaps the answer is not hiring a Black Doctor or Black Companion/Assistant, but engaging some Britons of African, Asian, or Middle Eastern familial descent in the writing, research and production pool.