Unable to download all hidden coordinates for my project

Not so obvious is how to download data with obscured localities.
I do have a project that allows me to do so, but it has over 445 000 observations and the only way I can get the obscured coordinates is by trying to download all 445 000 observations which always fails to finish. If I try any filtering for a taxon, then the obscured data does not download.
How can I download observations with the actual coordinates for hidden coordinates, without having to create new projects each time and getting special permission for each project from each of hundreds of users?

I split this report off from the feature announcement.