For me the fix was to install Individual Component on my Visual Studio 2022 Community:
Look for MSVC v143 VS 2022 C++ x64/x84 build tools (v14.36-17.6). I also deleted the one called (latest) but eventually installed it back.
After that I re-generated project files and reopened Unreal Engine, and Live Coding picked the right toolchain:
Using Visual Studio 2022 14.36.32545 toolchain