It should be possible to comply with LGPL without publishing the source code of the whole application. Either by running the application and ffmpeg in different isolates (wasm processes), or by offering a way to merge (link) the wasm code of the closed-source application with a user compiled FFmpeg wasm build.

Different isolates might even be enough to satisfy GPL, similar to how you can invoke FFmpeg as a command line tool from a closed application. Though that feels like legally shaky ground.