This is really, really neat. Never would have thought that Babel could be used for instrumentation, though after a bit of digging it looks like https://istanbul.js.org/ uses a similar trick!
Is there a way to integrate this into a local dev environment without needing to run the MITM proxy? (For instance, if you know you're loading everything through a single bundle.) Could be a really popular dev tool if so.
Also, have you considered using IndexedDB to store the logs client-side, further removing the need for a backend? Storage should be practically unlimited if you have free drive space on desktop Chrome.