Although large language models (LLMs) have shown promise in biomolecule optimization problems, they incur heavy computational costs and struggle to satisfy precise constraints. On the other hand, specialized solvers like LaMBO-2 offer efficiency and fine-grained control but require more domain expertise. Comparing these approaches is challenging due to expensive laboratory validation and inadequate synthetic benchmarks. We address this by introducing Ehrlich functions, a synthetic test suite that captures the geometric structure of biophysical sequence optimization problems. With prompting alone, off-the-shelf LLMs struggle to optimize Ehrlich functions. In response, we propose LLOME (Language Model Optimization with Margin Expectation), a bilevel optimization routine for online black-box optimization. When combined with a novel preference learning loss, we find LLOME can not only learn to solve some Ehrlich functions, but can even perform as well as or better than LaMBO-2 on moderately difficult Ehrlich variants. However, LLMs also exhibit some likelihood-reward miscalibration and struggle without explicit rewards. Our results indicate LLMs can occasionally provide significant benefits, but specialized solvers are still competitive and incur less overhead.