President Wilson also believed that blacks should not hold any position in government ... and he acted on his beliefs.
I don’t think either are true.
Lee suggested that he could teach black men to be soldiers, which would be a position in government.
Wilson segregated institutions such as the post office, but he certainly retained and recruited black soldiers in the army which would be a position in the government.
I would suggest that they would both be considered for some positions in government, and opposed to others.