At this point, I am curious if your previous nominations post type had the “has_archives” parameter set to true. This is because if it did, then with my experience there would very likely be conflicts with that page, though at times perhaps not realized.
If it’s not meant to have archives, then I think the act of visiting example.com/nominations/ wouldn’t run so much risk.
That said though, if you really want to get it imported in, you can use the Tools menu item and the “post types” tab to grab a copy of a JSON-version of the settings, and then edit a couple spots, then paste it in to import.
For example, at the start of that JSON I have for one of my personal sites, there’s this: {"music_video":{"name":"music_video","label":"Music Videos",...
If I wanted to rename that “music_video” slug to something else, say pluralization before importing, I could change it to: {"music_videos":{"name":"music_videos","label":"Music Videos",...
and then paste in the entirety of the JSON, on the left.
I’d say for you, maybe just get one post type registered so that there’s minimal issue with editing JSON data directly, and then after getting imported, proceed with using the UI for the rest of things.