LSJ tends to have more traditional styles, but are entirely hand made in the old tradition of the shoe maker. JLP, a subsidiary of Hermes, has more modern styles, but not sure their shoes are entirely hand made. JLP also has access to Hermes leather sources, so depending on what you are looking for either could be considered superior. For my money, LSJ is my preference.
Yes JLP bespoke shoes are entirely hand made in their Paris Mogador workshop. I remember an older report somewhere on the web showing their complete handmade process.
Maybe Lobb St James are known to make more traditional styles and they don't make any trial shoes for fitting but they are very flexible.