BERLIN—Criticism of Angela Merkel’s open-door refugee policy in Bavaria, the southern German state where most migrants enter the country, erupted on the German chancellor’s doorstep on Thursday in the shape of a busload of 31 refugees.

The bus had been chartered by the Bavarian town of Landshut in protest of what one local politician described as the municipality’s exhausted capacity to receive more migrants. Peter Dreier, a county…