Should I leave out research in my NSF fellowship application if I'm not getting an LOR from the professor I worked with?
As someone who has reviewed applications for the NSF Graduate Research Fellowship program, it would definitely be weird to see someone talk extensively about a research project and then not have the advisor write a supporting letter. It would certainly raise questions among the panelists assigned to review the application, especially if no other explanation were provided.

Normally I would not make such a suggestion, but this might be an instance where you should list the experience (if you can), but not talk about it so effusively or in great depth.