Hi Fredrik, I think I have a solution for you. I wish I knew more about the file structure and file naming for TBC but here goes.
So this morning I tried using your steps and drag-n-dropped a project.vce~ file into my v5 window. I made a change and tried to rename it with Save As. Like you say it's not allowed. Then I saved it and I now had a project.vce~~ file. I closed the project and did a drag-n-drop of the project.vce~~ file. Sure enough the changes I made were there. I changed a layer color and again saved the project. I now had a project.vce~~~ file. I closed that project.
In file explorer I did a right click on the project.vce~~~ file and selected Properties. The full name appeared in the top box. I renamed it by removing the ~~~ and adding a different name called project-test.vce and hit OK. A warning that it could cause problems popped up and I hit YES to change it. In TBC I chose File->Open and in my projects directory chose the project-test with the TBC logo. When the project opened it had all the changes I'd made.
I don't think we are supposed to use the .vce~ files because they do not show up when using the File->Open command. I personally do not use the drag-n-drop feature to add files because sometimes it has not worked correctly.
I hope this might work for you. I'm also wondering if some of the problems you are experiencing come from using Dropbox? I have not tried using a Cloud service for file storage yet. I have been using a Synology NAS device that gives me 12TB of networked storage. Recently though because v5 seems to run slower I have been keeping the current project files on my C drive and put a backup on the NAS. I'm looking forward to seeing the first v5.X update with hopes some of the issues being reported here are fixed.