This is most probably the same (or at least similar) to the issue people have with writing to Google Drive
https://forum.cockos.com/showthread.php?t=223414
My guess is that when the background dropbox code runs to upload the latest save, it temporarily locks the file and so Reaper cannot overwrite it. The larger the file, the more frequent this problem becomes as the time to upload to dropbox increases, hence the file is locked for longer time.
One strange thing is that you say that this happens also with new projects when you save them for the first time. Are you sure about this? If so, maybe the background dropbox code locks the whole folder while uploading files.
Maybe there is more useful stuff here
https://www.dropboxforum.com/t5/Drop...ox/td-p/325359