The pragmatist tenet of metaphysics takes the pluralist view that “there is more than one sound way to conceptualize the world and its content.”
And this pragmatist tenet of metaphysics seems similar to Bohr’s idea of “complementarity”. After the Solvay Conference, Bohr was noted to have remarked:
“I can quite understand why we cannot speak about the content of religion in an objectifying language. The fact that different religions try to express this content in quite distinct spiritual forms is no real objection. Perhaps we ought to look upon these different forms as complementary descriptions which, though they exclude one another, are needed to convey the rich possibilities flowing from man’s relationship with the central order.” 
Here Bohr openly endorses mutually exclusive explanations. Or, to put it more plainly, he openly endorses contradictions. Not only that, he suggests that contradictions are required in order to make sense of religion!
He was known to attempt to apply complementarity to far more than religion, most notably to quantum physics, as we have mentioned earlier.
So, he certainly believed that quantum physics required one to embrace a contradiction, that something could be both a wave and a particle, for example.
So it does seem like Bohr ascribed to pragmatist metaphysics as well.
Pragmatist philosophy of science holds the view that:
“a scientific concept or theory should be evaluated by how effectively it explains and predicts phenomena, as opposed to how accurately it describes objective reality.”
Bohr certainly believed that a scientific theory should not be judged on how accurately it described reality. This can be readily inferred from his rejection of the fact that we can know physical reality and instead should confine ourselves to the study of appearances.
And he was more than happy to make predictions based on quantum theory, often in terms of probabilities. That was the entire point of the probabilities, to allow some kind of (non-casual) prediction. As long as one did not claim that they were accurately describing a physical and objective reality.
Therefore, we can say that Bohr was indeed influenced by pragmatism.