Task-specific models have long dominated natural language processing tasks. However, general-purpose large language models (LLMs) have recently begun to successfully compete with highly specialized solutions across various NLP domains. In this paper, we investigate the applicability of LLMs to the task of estimating the semantic proximity of a word's meanings in a pair of usages, known as Word-in-Context (WiC). Drawing on the multilingual CoMeDi benchmark, we propose novel approaches to building automated WiC-systems based on LLMs. We conduct a systematic comparison of five different configurations in terms of quality and computational costs. In particular, we propose a configuration where LLM predictions are adjusted using a training set, without the need to fine-tune the LLM itself. Results on test sets across seven languages show that our proposed approaches enable LLMs to outperform all existing specialized systems, establishing a new state-of-the-art (SOTA) on the CoMeDi benchmark. Nevertheless, the achieved high quality comes with a significant increase in computational costs: LLM-based systems require several orders of magnitude more computations compared to compact specialized models (such as XL-DURel). This work represents a step towards understanding the trade-off between accuracy and resource efficiency when using modern LLMs in lexical semantics tasks.