This is super painful - I'm hoping that we can find a better process for this in the next pass at the QuickStart. You're definitely not alone in this pain, and thanks for reaching out!
I think the easiest way to do this will actually be to enable SSH into the QuickStart. If you go to VirtualBox, you want to go to:
Settings -> Network -> Advanced -> Port Forwarding
Then click the "+" icon, and add an entry for SSH that looks like the following. Mapping port 5001 on your host to port 22 on the guest.
Once you have this setup, you can ssh to the QuickStart in a normal terminal using something like:
ssh -p 5001 rstudio@localhost
Using PuTTy (for Windows users) will be similar - localhost is the host, rstudio is the user, and 5001 is the port.
This should allow you a better workflow for copying and pasting. Do you mind giving that a try and letting me know how it goes?