Have you tried making a different desktop entry/shortcut for each Firefox profile and then setting a browser picker as your default browser?

- Junction (Linux browser picker): https://github.com/sonnyp/Junction

- Finicky (macOS browser rule setter): https://github.com/johnste/finicky and Browserosaurus (macOS browser picker): https://github.com/will-stone/browserosaurus

- Hurl (Windows browser picker): https://github.com/U-C-S/Hurl