Why Britain might accept Russian control of Constantinople

Would Britain and France really let Russia keep Constantinople after their Crimean War experience?

Possibly, because holding Constantinople would consume Russian attention in the former Ottoman sphere and reduce pressure toward India.

Constantinople would be a long-term strategic burden as well as a prize. Russia would spend years managing hostility from the remnants of the Ottoman Empire, reducing its ability to threaten India. Britain would also use the interval to strengthen its position in Persia and other approaches to the subcontinent. The logic accepts continued Ottoman hostility as the price of redirecting Russian attention away from the Great Game.