I am looking for opinions of how to handle large binary files on which my source code (web application) is dependent. What are your experiences/thoughts regarding this?
I personally have run into synchronisation failures with Git with some of my cloud hosts once my web applications binary data notched above the 3 GB mark. I considered BFT Repo Cleaner at the time, but it felt like a hack. Since then I've begun to just keep files outside of Git purview, instead leveraging purpose-built tools such as Amazon S3 for managing files, versioning and back-up.
Does anybody have experience with multiple Git repositories and managing them in one project?
Yes. Hugo themes are primarily managed this way. It's a little kudgy, but it gets the job done.
My suggestion is to choose the right tool for the job. If it's for a company and you're managing your codeline on GitHub pay the money and use Git-LFS. Otherwise you could explore more creative options such as decentralized, encrypted file storage using blockchain.