Paclitaxel and Trastuzumab With or Without Lapatinib in Treating Patients With Stage II or Stage III Breast Cancer That Can Be Removed by Surgery

Inclusion Criteria: * Pathologic confirmation of invasive breast cancer; patients with inflammatory breast cancer are not eligible * Clinical stage II-III operable invasive breast cancer with intent to perform surgical resection after neoadjuvant therapy * Patients with multicentric or bilateral disease are eligible as long as the target lesion meets the eligibility criteria for this study * Staging to rule out metastatic disease is recommended for clinical stage III patients * Tumors must be HER2 positive defined as HER2 3+ by immunohistochemical (IHC) assays or gene amplification by fluorescence in situ hybridization (FISH) with a ratio of \>= 2 on invasive tumor * Estrogen receptor (ER) and progesterone receptor (PgR) status must be known * The target lesion in the breast must be \>= 1 cm on physical examination or by radiographic measurement; palpable axillary adenopathy will be documented but not serve as measurable disease for the primary endpoint; patients with axillary disease only are not eligible to participate * Patient agrees to provide pretreatment biopsies * No prior chemotherapy, hormone therapy, biologic, or radiation therapy with therapeutic intent for this cancer * Cardiac ejection fraction must be \>= 50% by echocardiogram or multiple gated acquisition (MUGA) scan * Eastern Cooperative Oncology Group (ECOG) (Zubrod) performance status 0-1 * Patients must not be pregnant or nursing * Absolute neutrophil count (ANC) \>= 1,000/ul * Platelet count \>= 100…