$ENV{PATH} is but one possible environment problem problem. Study %ENV for any differences and see if eliminating them makes the problem go away. For instance LD_LIBRARY_PATH controls finding dynamically loaded libraries on several versions of Unix - that can definitely affect whether lyx can run. It might also want to talk to an X server, which means that things like DISPLAY might be needed. And so on.

Also you don't say whether or not you looked in your server's error logs. If you haven't done so then replying to people here and getting further rounds of guesses is pretty pointless. We can think of a million things that could go wrong. The server's error logs give you feedback on what is wrong.