Thank you for the suggestions! I took a couple pictures to illustrate the issue, in case anybody else runs into the same thing. As you can hopefully see, if I tighten the washer to just be flush with the end of the rod, the rest of the emitter is loose and rests in a much lower position. If I screw the washer on all the way, some of the rod is visible.
Between the suggestions so far, I'm wondering which would result in the emitter sitting at the correct height. If I glue the parts, will the emitter sit too high? If I screw the rod in further, will the emitter sit too low? Or is there no difference because the gluing option would also result in the rod being screwed in further? (Reading Gregatron's post again, I think that might be the case.)
Looking at a couple reference photos of the prop, it seems like everything's in the right position in my second picture other than the rod, but I figured I'd check before doing anything hasty and messing up my new toy...