1) Install dropbox or equivalent cloud storage product and sync needed files that way between computers. Remember, you can allow only certain folders to be synced on specific devices (you don't have to sync the entire dropbox, just the folders you need)
2) If you are allowed to setup more than one user on the remote server, have a 2nd user and then have user2 session connect rdp session to user1. This will keep the user1's gui alive in the cloud without having to remain logged in to rdp locally.
This video should show you how to implement this 2 user setup on your server to hold an rdp session open. Note that this does 'permanently' use 1 rdp session until you decide to close it.
[markdown cannot embed video :( ]
Then use AmmyyAdmin AnyDesk on user1's desktop to connect and manipulate the desktop. This includes using AnyDesk's file manager's ability to browse any folder you need and copy. AnyDesk can be free if you connect via direct IP connection. Most vps servers have dedicated IP addresss or subdomain address so this should not be a problem. Good idea to password protect your AnyDesk login and which IDs have access to unattended remote connections. The AnyDesk file manager is a bit crude, but it works. Their big thing is simplicity and speed.
Note: Use portable mode only on the remote user's desktop; Do NOT fully install AnyDesk. Also, the CPU usage might increase to stream the desktop screen, somewhat related to the size of the RDP window. I am using 1280 x 2048 window with 4 cores and the CPU usage is 22-25% idle or moving things around. This might decrease if there is more video ram or graphics processor on the target server. But, if you only "browse files" (use only the file manager without streaming the desktop), CPU usage >0.3% idle and >1 avg% when transferring files (burst up to 5-6% when the file is finished uploading and the pieces are being finalized).
You'd have to write your own scripts (java, .net, c#/c++, AutoIT, etc) to launch AnyDesk locally and automate the connecting and downloading specific files.
This strategy is a bit more complex, but it should do the job. Not sure why microsoft rdp cannot have some simple, quick file manager like what ammyy admin AnyDesk has; oh well.
Add: Can also use AnyDesk or Teamviewer. Teamviewer became a lot more restrictive on what is considered to be "non commercial use", but Anydesk is secure, much smaller footprint, and if you can have a direct connection doesn't seem to care too much about usage. If you do need a license, it will be much lighter on the wallet.
AnyDesk works flawlessly without any installation required. In fact, if using in a server environment as I described above, no installation is recommended.
Edit: AmmyyAdmin is no longer recommended for several months now due to some security and technical concerns. Added AutoIT as a scripting capability to automate interaction with GUI/nearly any windows function.