VSC w/ ESP-IDF: how do I configure LSP to have autocompletion for bridged headers?

I asked a similar question here elsewhere in regards to another project that had a similar structure, albeit it uses makefiles instead of cmake.

TL:DR it appears that what we were trying was for SwiftPM project, which we are not using.

In your case, you appear to be using CMAKE, so you can seemingly, in fact, achieve autocomplete using CMAKE_EXPORT_COMPILE_COMMANDS to create the compilation database.

For those using makefiles, you apparently need to craft the compilation database json yourself.