We introduce RosettaSearch, an inference-time multi-objective optimization approach for backbone conditioned protein sequence design. We use large language models (LLMs) as a generative optimizer within a search algorithm capable of controlled exploration and exploitation, using rewards computed from RosettaFold3, a structure prediction model, under a strict computational budget. In a large-scale evaluation, we apply RosettaSearch to 400 suboptimal sequences generated by LigandMPNN (a state-of-the-art model trained for protein sequence design), recovering high-fidelity designs that LigandMPNN's single-pass decoding fails to produce. RosettaSearch's designs show improvements in structural fidelity metrics ranging between 18% to 68%, translating to a 2.5x improvement in design success rate. We observe that these gains in success rate are robust when RosettaSearch-designed sequences are evaluated with an independent structure prediction oracle (Chai-1) and generalize across two distinct LLM families (o4-mini and Gemini-3), with performance scaling consistently with reasoning capability. We further demonstrate that RosettaSearch improves the sequence fidelity of ProteinMPNN designs for de novo backbones from the Dayhoff atlas, showing that the approach generalizes beyond native protein structures to computationally generated backbones. We also demonstrate a multi-modal extension of RosettaSearch with vision-language models, where images of predicted protein structures are used as feedback to incorporate structural context to guide protein sequence generation. To our knowledge, this is the first large-scale demonstration that LLMs can serve as effective generative optimizers for backbone-conditioned protein sequence design, yielding systematic gains without any model retraining.