How did Teddy Roosevelt reconcile being both a progressive and an imperialist as President?
Virtually every President since Andrew Johnson up until Teddy Roosevelt had been elected either through Gilded Age corruption and influence-peddling or military fame. Roosevelt, by contrast, became President because Leon Czolgosz murdered his boss, President McKinley, therefore he was beholden to neither the progressives nor the robber barons of his day, and could craft a presidency based on his personality and beliefs.
Part of that personality and belief keenly followed his sense of adventure, hard work and opportunity. Roosevelt didn't mind big business, he just wanted consumers to be safer and the competition to be reasonably fair. He loved the outdoors and was thus a natural conservationist. In short, Teddy Roosevelt was a progressive because he really was progressive.
It wasn't as hard for him (or anyone living in that era) to reconcile imperialism with progressivism. The same sense of opportunity and exploration that governed TR's ideals domestically drove at least part of his imperialist foreign policy. He also wholeheartedly believed that America was destined for the world stage in Latin America and Asia, so in that sense, he was imperialist because he was very patriotic and a bit nationalist.
Roosevelt was somewhat progressive on race domestically. He had desegregated New York's schools when he was Governor and he persuaded San Francisco to allow Japanese students into whites only schools. He had blacks on his Cabinet and appointed the country's first Jewish cabinet minister. But that doesn't mean he ardently pushed for civil rights equality--he didn't, and racism was a factor in his support of an American military presence in Panama and the Caribbean, as well as a continued war to subjugate the Philippines. This partly explains the apparent contradiction between progressivism and imperialism.