TL;DR - Is something broken on the "Get Movie & TV Info..." function used in Auto Import. It isn't consistent with a manual run for multiple media files, but that allows overwrite of values if you tick the box for that. The name found on TMDB isn't saved to the [name] tag as expected.
Your story is complicated a bit Jens because you are renaming automatically somehow, hopefully externally to MC, and haven't shared examples of exact directory and file names that result.
Then the short name version that you search for matches to the wrong record on TMDB.
Then because you are running the "Get Movie & TV Info..." function within Auto Import, I believe MC won't overwrite existing tags, and the [name] tag has already been created by the import process when the "Get Movie & TV Info..." function runs.
Anyway, just for giggles I created a dummy video, named it "\Star Wars Episode 3\Star Wars Episode 3.mp4", and placed the sub-directory in a directory watched by Auto Import in MC.
The first time I did this I used a small file. MC didn't recognise it as a Movie and left the [Media Sub Type] tag blank. So naturally MC didn't run the "Get Movie & TV Info..." function against it. When I tried to manually run the "Get Movie & TV Info..." function against the file, I received an information message from MC telling me the [Media Sub Type] tag must be "Movie" or "TV Show". Is that what you see? Do you see that message when you first try to run the "Get Movie & TV Info..." function on one of these files?
Then it got interesting.
I found a larger video file that was not tagged as a movie, renamed it as above to "\Star Wars Episode 3\Star Wars Episode 3.mp4", and placed the sub-directory in a directory watched by Auto Import in MC.
It imported, and the "Get Movie & TV Info..." function ran, collecting meta data for the movie, plus Cover Art. But the [name] tag stayed the same as I had named the file, "Star Wars Episode 3", just as you had said.
Then I noticed, this wasn't the meta data for "Star Wars: Episode III - Revenge of the Sith" but was, in fact, the meta data for another program found via the lookup of "Star Wars Episode 3". The wrong "Movie" had been found by the "Get Movie & TV Info..." function! It was "Star Wars Knights of the Old Republic: Episode 3 - The Circle of Fate", and was the first match on TMDB when I ran the "Get Movie & TV Info..." function manually.
When I saved the meta data found on TMDB, the [name] tag was changed to the name for this first match found on TMDB, "Star Wars Knights of the Old Republic: Episode 3 - The Circle of Fate", just as you had seem with your testing.
So, the "Get Movie & TV Info..." function run with Auto Import would have gotten the incorrect data from TMDB, but, you are correct, the [name] tag is not saved when Auto Import runs the "Get Movie & TV Info..." function.
I think what is happening is that the "Get Movie & TV Info..." function used in Auto Import is not allowed to overwrite existing tag data, while manually running the "Get Movie & TV Info..." function against one media file defaults to overwriting tags, and manually running it against multiple files allows you to select whether tags are overwritten. If that is the case, what you are seeing is by design. I can sort of understand why that is done, as in your Star Wars example, the wrong match can be found on TMDB, and if that happens all existing tag data would be lost if the function defaulted to overwriting tags.
I started the test process above again with the "Get Movie & TV Info..." function turned off in Auto Import. Cover Art wasn't collected, and neither was any data for the movie.
I selected two movies to run the "Get Movie & TV Info..." function against, so that the selection of the correct movies would be done automatically in a similar way to when it is run via Auto Import. The process completed, selecting the wrong movie for Star Wars Episode 3 again, but this time the [name] tag was saved correctly for the "Star Wars Knights of the Old Republic: Episode 3 - The Circle of Fate" record found. Of course, I had the "Allow overwrite of existing values" check box ticked.
Is this by design JRiver? I certainly expected MC to save at least the name found on TMDB to the [name] tag, regardless of overwriting rules. But then I rarely import movies using Auto Import, without following up with a manual "Get Movie & TV Info..." function. The exception is TV recordings, and that process doesn't run the "Get Movie & TV Info..." function at all, so I have to manually run the "Get Movie & TV Info..." function.