Senate Republicans on Tuesday turned to Miguel Estrada, a man whom Democrats denied a presidential appointment in 2003, to be their lawyer as they fight to overturn some of President Obama’s own appointments. The move escalates the simmering constitutional battle between the Congress and the White House over several appointments Mr. Obama made in January, when the Senate was still meeting every few days and considered itself in session. Senate Minority Leader Mitch McConnell, Kentucky Republican, said he and his colleagues are filing friend-of-the-court briefs in an ongoing challenge to Mr. Obama’s appointments, and said Mr. Estrada will be their...