Research software metadata can be found across many code repositories and software registries. Here, we describe the tooling for a bidirectional bridge between the software development platform GitHub and the ELIXIR bio.tools registry of life sciences software tools and data resources. The developed bridge maps and improves metadata records across these two platforms, thereby benefiting both and helping make research software more FAIR: findable, accessible, interoperable, and reusable. Specifically, the bridge enables production of high-quality, rich bio.tools entries from the content already available in GitHub repositories, and uses bio.tools records to suggest improvements to GitHub repositories through pull requests or issues. This includes adding missing information and standardized descriptions for increased compliance with Software Management Plans. The bidirectional bridge makes extensive use of existing APIs (GitHub, bio.tools, Europe PMC) and large language models (LLMs) to enrich metadata on both platforms. By automating metadata extraction, improvement suggestion, and integration, the bridge reduces the manual overhead required to FAIRify research software, lowering barriers for researchers to contribute or maintain well-annotated, reusable software.