This phase II trial tests how well tebentafusp works to shrink tumors prior to primary treatment with surgery or radiation in patients with uveal (eye) melanoma that has spread to nearby tissue or lymph nodes (locally advanced) or that cannot be removed by surgery (unresectable). Tebentafusp is a drug that binds to melanoma tumor cells as well as immune cells called T-cells. This binding causes an immune response against the melanoma cells, which leads to tumor cell death. Tebentafusp has been approved for the treatment of locally advanced and unresectable uveal melanoma. Giving tebentafusp before primary treatment with surgery or radiation may help shrink the tumor, prevent the disease from spreading, or reduce the likelihood that patients will require total eye removal (called enucleation).