I have put together an autosplitter for Neon White: https://github.com/crashb/neon-white-autosplitter
The autosplitter works with all of the Level Rushes. For details, please see the autosplitter's readme file.
I did briefly investigate the possibility of removing load times in the autosplitter, but I was not able to figure it out quickly. Once the game is fully released I can look into adding this functionality to the autosplitter. I agree that load removal would be a big improvement.
I would prefer not to have to start a new file - skipping through cutscenes and watching level completion animations adds a lot of tedium to the run. If you play on a completed save then you can just go through all the levels in order.
RTA should definitely be used instead of IGT to prevent excessive pausing during gameplay.
I put together an autosplitter to be used in an "All Main Missions" category. It uses RTA, so it includes time spent in the menus. See the readme on GitHub for more details.
Autosplitter link: https://github.com/crashb/neon-white-autosplitter
EDIT: You can now download the autosplitter directly through LiveSplit without having to visit GitHub. Right-click the LiveSplit timer, then click "Edit Splits..." then select "Neon White" in the first dropdown.