This randomized phase II trial studies capecitabine and lapatinib ditosylate to see how well they work compared with capecitabine, lapatinib ditosylate, and cixutumumab in treating patients with previously treated human epidermal growth factor receptor 2 (HER2)-positive stage IIIB-IV breast cancer. Drugs used in chemotherapy, such as capecitabine, work in different ways to stop the growth of tumor cells, either by killing the cells, by stopping them from dividing, or by stopping them from spreading. Lapatinib ditosylate may stop the growth of tumor cells by blocking some of the enzymes needed for cell growth. Monoclonal antibodies, such as cixutumumab, can block the ability of tumor cells to grow and spread. It is not yet known whether capecitabine and lapatinib ditosylate are more effective when given with or without cixutumumab in treating breast cancer that has spread nearby or to other areas of the body.