Good answer. Asking an AI coding agent to port a C codebase to Rust is clearly and obviously creating a derivative work. You need to follow the terms of the LGPL license, and your Rust fork of FFMPEG must be licensed as LGPL. Videolan holds the copyright to FFMPEG, and that includes derivative works. LGPL allows you to do exactly what you're doing.
Note that if you convert x264, x265 or any of the other GPL v2 libraries, your Rust version of those libraries will be GPL v2. When builds of FFMPEG are created with GPL v2 libraries, the entire build becomes GPL v2 (which has a more restrictive copyleft provision than LGPL).
Good to know! I have licensed as LGPL so I think that covers it. Rewriting x264 and x265 are in scope so I’ll keep that in mind. I will design the application as well so that those can easily be excluded from compilation to get a pure LGPL build.